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Admixture and Drift in Oscillating FluidFlows

By V. A. V l a d im i r o v

Dept of Mathematics, University of York, Heslington, York, YO10 5DD, UK

(Received September 21st, 2010)

The motions of a passive scalar a in a general high-frequency oscillating flow are studied.Our aim is threefold: (i) to obtain different classes of general solutions; (ii) to identify,classify, and develop related asymptotic procedures; and (iii) to study the notion of driftmotion and the limits of its applicability. The used mathematical approach combines aversion of the two-timing method, the Eulerian averaging procedure, and several novelelements. Our main results are: (i) the scaling procedure produces two independent di-mensionless scaling parameters: inverse frequency 1/ω and displacement amplitude δ; (ii)we propose the inspection procedure that allows to find the natural functional forms ofasymptotic solutions for 1/ω → 0, δ → 0 and leads to the key notions of critical, sub-critical, and supercritical asymptotic families of solutions; (iii) we solve the asymptoticproblems for an arbitrary given oscillating flow and any initial data for a; (iv) thesesolutions show that there are at least three different drift velocities which correspond todifferent asymptotic paths on the plane (1/ω, δ); each velocity has dimensionless mag-nitude O(1); (v) the obtained solutions also show that the averaged motion of a scalarrepresents a pure drift for the zeroth and first approximations and a drift combined withpseudo-diffusion for the second approximation; (vi) we have shown how the changing ofa time-scale produces new classes of solutions; (vii) we develop the two-timing theoriesof a drift based on both the GLM -theory and the dynamical systems approach; (viii)examples illustrating different options of drifts and pseudo-diffusion are presented.

1. Introduction

The transport of a scalar field a by an oscillating velocity field u represents a classicalproblem of fluid dynamics. One can find a number of models, physical effects, and applica-tions in Taylor (1953), Moffatt (1983), Hydon & Pedley (1993), Frisch (1995), Moffatt (1998),Magar & Pedley (2005), Leal (2007). The transport of a scalar represents a paradigmproblem for the use of the dynamical systems approach in fluid dynamics (see Aref (1984),Ottino (1989), Solomon, Lee, and Fogelman (2001), Samelson & Wiggins (2006)) and forthe studies of turbulence (see Sreenivasan (1991), Warhaft (2000), Balk (2002)). It iswell-known that even in purely oscillating flows the motion of a material particle con-sists of two parts: oscillating and non-oscillating. The latter is known as a drift, seeStokes (1847), Maxwell (1870), Lamb (1932), Longuet-Higgins (1953), Darwin (1953), Hunt (1964),Batchelor (1967), Schlichting (1979), Lighthill (1978a), Andrews & McIntyre (1978), Craik (1982),Grimshaw (1984), Craik (1985), Benjamin (1986), Eames & McIntyre (1999), Eames, Belcher, and Hunt (1994).Different notions of a drift are used in physics, see e.g. Vergassola & Avellaneda (1997),Chierchia & Gallavotti (1994).In our paper we consider a high-frequency asymptotic problem of a scalar trans-

port by a given oscillatory velocity field. A characteristic length L for both u and

2 V. A. Vladimirov

a is arbitrary. We use a simple and straightforward formulation of the problem thatallows to study the general properties of solutions without the consideration of par-ticular applications. The evolution of a is exploited for the study of the concept of adrift motion and the limits of its applicability; this approach allows a great degree offlexibility and has not been used before. We employ the mathematical method whichcomprises the two-timing approach (Nayfeh (1973)), the fast-time averaging operation,and several novel elements; we call it two-timing-and-averaging method (TTAM). Theinterplay of the two-timing form of solutions and a fast-time-averaging operation wasexploited by many authors. The well-known applications in classical mechanics belongto Kapitza (1951a), Kapitza (1951b), Landau & Lifshits (2000). A simplified version ofthe two-timing method has been extensively used in mechanical engineering and isknown as vibrational mechanics, see e.g. Blekhman (2000), Blekhman (2004). HoweverKapitza (1951a), Kapitza (1951b), Landau & Lifshits (2000), Blekhman (2000), Blekhman (2004)did not use (and even deliberately denied) the underlying asymptotic nature of themethod. This nature has been emphasised and exploited by Vladimirov (2005), Yudovich (2006),Vladimirov (2008) in a particular form of TTAM used in this paper.Let us outline here the content and the main results:In Sect.2 we formulate the problem, introduce two-timing and dimensionless versions

of the governing PDE. The two independent dimensionless scaling parameters are inversefrequency 1/ω and displacement amplitude δ of oscillations. Our asymptotic procedureimplies both 1/ω → 0 and δ → 0. It produces the multiplicity of available asymptoticpaths on the plane (1/ω, δ); we parameterize these paths with a single small parameterεα = 1/ωα, α = const > 0. This multiplicity gives us new opportunities for the studyof a drift. At the end of the section we consider an important case when u does notcontain any slow time-scale T : in this case we show the presence of an infinite number ofsolutions with a range of different T .In Sect.3 we develop the inspection procedure for the governing equations. It leads to

classification of different asymptotic solutions and to the key notions of critical, sub-critical, and super-critical asymptotic families of solutions. For a critical asymptoticfamily: (i) the small parameter is ε1/2 (or α = 1/2); and (ii) the correlation 〈ua〉 betweenthe oscillatory parts u and a of u and a enters the O(1) approximation of the averagedgoverning equation. For a super-critical family α < 1/2 and again 〈ua〉 = O(1), whilesome constrains on u must be imposed. For a sub-critical asymptotic family 〈ua〉 appearsonly in higher than O(1) approximations.In Sect.4 and Appendix A we obtain the general solution of an asymptotic problem for

the critical asymptotic family (α = 1/2) with an arbitrary purely oscillating velocity u

and arbitrary initial conditions for a. Each solution a = a+ a consists of explicit formulaefor its oscillating part a and an advection-diffusion-type equation for the averaged parta. Our detailed analytical calculations include five steps of successive approximations.The obtained solutions satisfy the governing equations with small explicitly calculatedresidual right-hand-sides. It appears that evolution of the averaged field a includes adrift with the velocity V = V 0 + εV 1 + ε2V 2 + O(ε3) and pseudo-diffusion with amatrix of coefficients κik = ε2χik + O(ε3), where ε ≡ ε1/2 and V 0,V 1,V 2, χik havemagnitudes O(1). Our term pseudo-diffusion means that the evolution of a is describedby an advection-diffusion-type equation where the diffusion-type term appears not in theleading approximation.In Sect.5 and Appendix B we present the asymptotic solutions for supercritical asymp-

totic families with α = 1/3 and 1/4 and degenerated velocity field u. For α = 1/3 the re-quired degeneration is V 0 ≡ 0 which leads to V = V 1+εV 2+O(ε2) and κik ≡ 0+O(ε2),where ε ≡ ε1/3. For α = 1/4 the required degeneration is V 0 ≡ 0 and V 1 ≡ 0 which leads

Admixture and Drift in Oscillating Fluid Flows 3

to V = V 2 +O(ε) and κik ≡ 0+O(ε), where ε ≡ ε1/4. Hence, the vanishing of the main

term of a drift velocity V 0 ≡ 0 (or the first two terms V 0 ≡ 0 and V 1 ≡ 0) does not leadto the smallness of a dimensionless drift. Instead, a different expression V = O(1) appearsfrom a different asymptotic procedure. At the same time any supercritical oscillations donot produce pseudo-diffusion (within the precision of our calculations).

In Sect.6 we generalise the results of Sects.4 and 5 considering more general u, whichrepresents a power series of a small parameter ε and contains both oscillating and non-oscillating parts. The expressions for drift velocities appear to be much more general,while pseudo-diffusion stays the same as in Sect.4.

Sect.7 contains eleven examples of drift velocities and/or pseudo-diffusivity calculatedfor various flows. These examples are: (1) the velocity field u consisting of two arbitrarymodulated fields of the same frequency; (2) u that represents the Fourier series of mod-ulated fields; (3) the Stokes drift, which allows us to make comparison (in Sect.10 ) withclassical results; (4) u is a spherical ‘acoustic’ wave; (5) V 1-drift (Sect.5 ), which takesplace for V 0 ≡ 0; (6) u is a plane travelling wave of a general shape or the superpo-sition of two such waves; (7) the polynomial velocity u, which has a nonzero averageand represents polynomial of a small parameter; (8) u as the Bjorknes configuration oftwo pulsating point sources; (9) a simple u that produces a drift V 0 leading to chaoticdynamics of material particles; (10) the rigid-body-type oscillations of a fluid where apseudo-diffusion term in the equations appears naturally; this example explains bothmathematical and physical nature of pseudo-diffusion; (11) we consider a sub-criticalasymptotic procedure and show that the averaged equations for a contain pseudo-driftand pseudo-diffusion; it emphasizes the significance of the critical and super-critical pro-cedures.

The TTAM used in Sects.3-7 combines five components: (i) a version of the two-timing method; (ii) the averaging over fast-time; (iii) the inspection procedure with themultiplicity of asymptotic procedures; (iv) the obtaining of general asymptotic solutionsfor these procedures; and (v) the estimating of RHS-residuals (which has been onlymentioned in Sects.4-6 and is actually given in Appendices A,B).

In Sect.8 we study the links between the TTAM -results of Sects.2-7 and other theories:(i) the dynamical systems approach, see Aref (1984), Ottino (1989), Samelson & Wiggins (2006));(ii)GLM -theory by Andrews & McIntyre (1978); (iii) the classical drift theory by Stokes (1847),Longuet-Higgins (1953), Batchelor (1967); (iv) the Krylov-Bogoliubov averaging method,see Bogoliubov & Mitropolskii (1961), Krylov & Bogoliubov (1947), Sanders & Verhulst (1985);(v) the homogenization method, see Bensoussan, Lions and Papanicolaou (1978), Berdichevsky, Jikov, and Papanikolaou (1999);(vi) the theory with nonzero molecular diffusion; and (vii) the MHD-kinematics in oscil-lating flows, see Moffatt (1978). The links between our TTAM -results of Sects.2-7 andthe theories (i), (ii) can be established only if we adapt and develop last two theories.Therefore, in Sect.8.1 we calculate a drift velocity by the averaging of characteristics fortransport equations and in Sect.8.2 we develop a two-timing version of GLM -kinematics.It is important that the TTAM -theory of Sects.2-7 uses the Eulerian averaging whilethe theories (i) and (ii) operate with the Lagrangian averaging. These different averagingoperations lead to different results for a drift and pseudo-diffusion.

Finally, in Sect.9 we discuss the assumptions used, the results obtained, and the prob-lems to be addressed. We also present short discussion/remarks at the end of each sectionand subsection.

4 V. A. Vladimirov

2. Basic Equations and Operations

2.1. Exact problem

The equation for a scalar field a = a(x∗, s∗) in Cartesian coordinates x∗ = (x∗1, x

∗2, x

∗3)

and time s∗ is(

∂

∂s∗+ (u∗ · ∇∗)

)a(x∗, s∗) = 0 (2.1)

where u∗ = u∗(x∗, s∗) is a given velocity field, the asterisks mark dimensional variables.All functions used in this paper are considered to be sufficiently smooth. Equation (2.1)describes the motion of a Lagrangian marker in either an incompressible or compressiblefluid or the advection of a passive scalar admixture with concentration a(x∗, s∗) in anincompressible fluid. For a compressible fluid it also describes the advection of a passivescalar admixture, where a represents the ratio of concentration of admixture to densityof a fluid. We study the motion of a Lagrangian marker (2.1) due to our primary interestin the motions of material particles.The velocity field u∗ has the functional form of a hat-function

u∗ = u∗(x∗, t∗, τ); t∗ = s∗, τ = ω∗s∗ (2.2)

where t∗ and τ are two mutually dependent time variables, which are introduced as twodifferent time-scales k1s

∗ and k2s∗ with k1 = 1 and k2 = ω∗ (ω∗ is frequency). The τ -

dependence in (2.2) is always 2π-periodic. The functional form (2.2) is aimed to describemodulated oscillations of high-frequency. Following established terminology we call t∗

slow time and τ fast time. We do not require (2.2) to satisfy any equations of motion;such a general setting is often accepted, for example in the kinematic MHD-dynamotheory (Moffatt (1978)).The functional structure of u∗ (2.2) underpins the idea that the solution of (2.1) also

represents a hat-function:

a = a(x∗, t∗, τ) (2.3)

Then after the use of the chain rule the equation (2.1) is

D∗a ≡(ω∗ ∂

∂τ+

∂

∂t∗+ (u∗ · ∇∗)

)a = 0 (2.4)

∂

∂s∗= ω∗

∂

∂τ

∣∣∣∣x∗,t∗

+∂

∂t∗

∣∣∣∣x∗,τ

≡ ω∗∂

∂τ+

∂

∂t∗(2.5)

We also denote partial derivatives by subscripts, e.g. aτ ≡ ∂a/∂τ .We obtain the dimensionless form of (2.4) by introducing the characteristic scales of

slow time T , length L, and velocity U . An example of a velocity field is

u∗ = U(1 + sin τ)(1 + C sin(t∗/T )) g(x∗/L, y∗/L, y∗/L) (2.6)

where g is an arbitrary vector-function of three scalar variables (one can also take T =L/U , but it narrows the class of velocity fields). Dimensionless variables and parametersare not asteriated:

t = t∗/T, x = x∗/L, u = u∗/U, ω = ω∗T (2.7)

According to Buckingham’s π-theorem the problem (2.4)-(2.7) possesses two independentdimensionless scaling parameters

ε1 ≡ 1/ω∗T = 1/ω, δ ≡ U/(ω∗L) (2.8)

Admixture and Drift in Oscillating Fluid Flows 5

where δ can be interpreted as either of two physical parameters: (i) characteristic dimen-sionless displacement δ = δ∗/L (for displacement δ∗ ≡ U/ω∗); or (ii) Strouhal numberδ = Ω/ω∗ (for frequency Ω ≡ U/L). The parameter ε1 represents another Strouhalnumber and follows a more general notation:

εα ≡ 1/ωα, with α = const > 0. (2.9)

The dimensionless versions of (2.5), (2.4) are

Da ≡(ω

∂

∂τ+

∂

∂t+ ωδu · ∇

)a = 0 (2.10)

∂

∂s= ω

∂

∂τ

∣∣∣∣x,t

+∂

∂t

∣∣∣∣x,τ

≡ ω∂

∂τ+

∂

∂t(2.11)

Eqn. (2.10) can also be written in the form containing only small parameters:

Da/ω = aτ + ε1at + δu · ∇a = 0 (2.12)

One can see that we operate on the plane (εα, δ) of two independent dimensionless scalingparameters. To use a rigorous asymptotic procedure one has to choose a one-parametricasymptotic path on this plane. In order to form such a path one might prescribe thedependence of each characteristic parameter in (2.7) on the chosen single parameter εα.The simplest choice is

T = const, L = const, UT/L = O(εα) (2.13)

We will exploit different options for paths, which go to the same limit

(εα, δ(εα)) → 0 as εα → 0 (2.14)

Remark: More details on the scaling procedure are given in Sect.2.3 and Sect.7.3.

2.2. The classes of H,B,T, and O(1)-functions

Notations and definitions:Definition 1. The class H of hat-functions is defined as

f ∈ H : f(x, t, τ) = f(x, t, τ + 2π) (2.15)

where t = s and τ ≡ ωs are two mutually dependent time variables (ω∗ is dimensionlessfrequency); the τ -dependence is always 2π-periodic; the dependencies on x and t are notspecified.Comments: (A) We have already accepted that u∗ ∈ H (2.2) and a ∈ H (2.3). (B) For

any f ∈ H a partial time-derivative can be expressed via the chain rule as

∂f

∂s=

(∂

∂t+ ω

∂

∂τ

)f(x, t, τ) ≡ ft + ωfτ , where t = s, τ = ωs (2.16)

(C) In any version of the two-timing method the variables t and τ are considered to bemutually independent after the chain rule (2.16) has been applied. The justification ofthis auxiliary assumption is given a posteriori (see Appendix A).

Definition 2. For an arbitrary f ∈ H the Eulerian averaging operation is

〈f〉 ≡ 1

2π

∫ τ0+2π

τ0

f(x, t, τ) dτ, ∀ τ0 (2.17)

where during the integration t = const and 〈f〉 does not depend on τ0.

6 V. A. Vladimirov

Definition 3. The class T of tilde-functions is such that

f ∈ T : f(x, t, τ) = f(x, t, τ + 2π), with 〈f〉 = 0, (2.18)

Comments: (A) One can see that a T-function represents a special case of H-function(2.15) with zero average (2.17). (B) In the text the tilde-functions are also called thepurely oscillating functions.Definition 4. The class B of bar-functions is defined as

f ∈ B : fτ ≡ 0, f(x, t) = 〈f(x, t)〉 (2.19)

Comments: (A) Any B-function depends on x and t and does not depend on τ ; inparticular, B-function can appear from H-function after its averaging. (B) With the useof the average (2.17) any H-function can be uniquely separated into its B- and T- parts:

f(x, t, τ) = f(x, t) + f(x, t, τ) with 〈f(x, t, τ)〉 ≡ f(x, t) (2.20)

Definition 5. The T-integration: For a given f we introduce a new function f τ called the

T-integral of f :

f τ ≡∫ τ

0

f(x, t, σ)dσ − 1

2π

∫ 2π

0

(∫ µ

0

f(x, t, σ)dσ)dµ. (2.21)

which represents the unique solution of the partial differential equation ∂f τ/∂τ = f with

the condition 〈f 〉 = 〈f τ 〉 = 0 (2.18).Comments: (A) The τ -derivative of T-function always represents T-function. However

the τ -integration of T-function can produce an H-function. An example: f = f1 sin τ

where f1 is an arbitrary function of x, t: one can see that 〈f〉 ≡ 0, however 〈∫ µ

0 fdτ〉 =f1 6= 0, unless f1 ≡ 0. Formula (2.21) keeps the result of integration inside the T-class.(B) The T-integration is inverse to the τ -differentiation

(f τ )τ = (fτ )τ = f . (2.22)

The proof is omitted.Definition 6. A dimensionless function f = f(x, t, τ, ε) (ε is a small parameter) belongs

to the class O(1) if f = O(1) and all partial x-, t-, and τ -derivatives of f (required forfurther consideration) are also O(1).Comments: In all text below: (A) The highest spatial derivatives will be of the fourthorder, while all t- and τ -derivatives will be of the first order; (B) All large or small param-eters (in all this paper) are represented by various degrees of ω only; these parametersappear as explicit multipliers in all formulae containing tilde- and bar-functions; thesefunctions always belong to O(1)-class.Definition 7. The commutator of two vector fields a(x) and b(x) is

[a, b] ≡ (b · ∇)a− (a · ∇)b. (2.23)

Comment: The commutator is antisymmetric and always satisfies Jacobi’s identity forthree vector fields a(x), b(x), and c(x):

[a, b] = −[b,a], [a, [b, c]] + [c, [a, b]] + [b, [c,a]] = 0 (2.24)

Two more useful properties of commutators are:

diva = 0, div b = 0 ⇒ div [a, b] = 0 (2.25)

a · n = 0, b · n = 0 ⇒ [a, b] · n = 0 (2.26)

Admixture and Drift in Oscillating Fluid Flows 7

where n is a normal vector to a surface ∂Ω.Properties of τ-differentiation and T-integration:For τ -derivatives it is clear that

fτ = fτ + fτ = fτ , 〈fτ 〉 = 〈fτ 〉 = 0 (2.27)

The product of two T-functions f and g forms a H-function: f g ≡ F , say. SeparatingT-part F from F we write

F = F − 〈F 〉 = f g − 〈f g〉 = ˜f g ≡ f g (2.28)

Since we do not use a two-level tilde notation for the T-parts of long expressions, we in-troduce braces instead. As the average operation (2.17) is proportional to the integration

over τ , for products containing functions f , g, h and their derivatives we have

〈f gτ 〉 = 〈(f g)τ 〉 − 〈fτ g〉 = −〈fτ g〉 = −〈fτ g〉 (2.29)

〈fτ gh〉+ 〈f gτ h〉+ 〈f ghτ 〉 = 0 (2.30)

〈f gτ 〉 = 〈(f τ gτ )τ 〉 − 〈f τ g〉 = −〈f τ g〉 = −〈f τ g〉 (2.31)

which represent different versions of the integration by parts. Similarly

〈[a, bτ ]〉 = −〈[aτ , b]〉 = −〈[aτ , b]〉, 〈[a, bτ ]〉 = −〈[aτ , b]〉 = −〈[aτ , b]〉 (2.32)

The equation to be solved:Let us accept that a dimensionless velocity in (2.10), (2.12) is

u(x, t, τ) ∈ H ∩O(1), and δ(ω) = δ0ωβ−1, β = const < 1 (2.33)

where u is a given function; indefinite constant β allows us to consider different asymp-totic paths; one can take δ0 ≡ 1 since any other δ0 can be incorporated into u. Eqn.(2.10) can be rewritten as

Da = ωaτ + at + ωβ(u · ∇)a = 0. (2.34)

In Sects.3-7 we will study and solve this equation.Remark: The τ -periodicity of the given velocity (2.2),(2.33) represents a restriction

which is artificially imposed to simplify further calculations. At the same time we willshow that a τ -periodic asymptotic solution (2.3) can be build for any initial data; hencewe are not forced to go outside the H-class (2.15).

2.3. Choice of scale T for t-independent velocity

A t-independent velocity

u = u(x, τ) (2.35)

is important theoretically. This case represents a degeneration, where the presence ofslow time t in eqns.(2.10),(2.34) is provided only by the additional agreement at ∈ O(1)leading to two mutually dependent time-scales (t, τ) = (s, ωs). It is clear that for (2.35)this agreement represents only one available option and an infinite number of differentslow-time-scales can be introduced. To make it possible one can write

τ = ω1−λ(ωλs) = ωλtλ ≡ τλ where ωλ ≡ ω1−λ, tλ ≡ ωλs, λ = const < 1− λ0(2.36)

with a small constant λ0 > 0 which provides ωλ → ∞ when ω → ∞. These manipulationsallow to introduce time-scales (tλ, τλ) = (ωλs, ωs) and a new version of eqn.(2.34)

(∂

∂s+ ωβuλ · ∇

)a = 0,

∂

∂s= ωλ ∂

∂tλ+ ω

∂

∂τλ(2.37)

8 V. A. Vladimirov

with uλ ≡ u(x, τλ). Further transformations of (2.37) yield(

∂

∂tλ+ ωλ

∂

∂τλ

)a+ ωβλ

λ (uλ · ∇)a = 0, (2.38)

where βλ ≡ (β − λ)/(1 − λ) and a = a(x, tλ, τλ). Finally, we replace the old scalingagreement at ∈ O(1) with a new one atλ ∈ O(1) (aτλ ∈ O(1) remains valid since τλ = τ).Now one can see that the problem (2.38) is mathematically identical to (2.34), whichmeans that for any solution in variables (t, τ) we have an additional solution in variables(tλ, τλ) (and vice versa); the varying of λ gives us an infinite number of such additionalsolutions. Hence, an infinite number of new solutions to (2.38) can be obtained by rescal-ing of the known solutions to (2.34), which will be considered in Sects.3-7. The existenceof such additional solutions means that motions with any slow-time-scale ωλt (2.36) arepossible. Indeed, it sounds physically reasonable: if a slow-time-scale is not enforced ex-ternally (does not appear in the given velocity) then any time-scale that is longer thanimposed oscillations can be considered as a slow one. For brevity, in the text below wekeep at ∈ O(1) and two time-scales (t, τ) = (s, ωs).Remark: Similar to (2.35),(2.38), the multiplicity of slow-time-scales takes place, for

instance, in dynamics of a classical pendulum with a harmonically vibrating pivot, whenan external gravity field is absent.

3. The Inspection Procedure

For simplicity we consider now eqns.(2.33),(2.34) with purely oscillatory velocity u =u ∈ T ∩O(1). In our inspection procedure we use a test solution to (2.34)

a(x, t, τ) = a(x, t) +1

ωαb(x, t, τ); α = const > 0 (3.1)

where a ∈ B ∩ O(1), b ∈ T ∩ O(1) and the amplitude of an oscillating part is given bythe small parameter εα = 1/ωα (2.9). The substitution of (3.1) into (2.34) yields

at +1

ωα

(ωbτ + bt

)+ ωβ

((u · ∇)a+

1

ωα(u · ∇)b

)= 0 (3.2)

The B-part of (3.2) is

at + ωβ−α〈(u · ∇)b〉 = 0 (3.3)

and the subtraction of (3.3) from (3.2) produces the T-part of (3.2)

ω1−αbτ + ω−αbt + ωβ(u · ∇)a+ ωβ−α(u · ∇)b = 0 (3.4)

with the notation · (2.28) used. In each underlined group in (3.4) the first term domi-nates over the second one (due to α > 0 (3.1)). Therefore, to make this equation mean-ingful we need to accept that the dominating terms are of the same order:

1− α = β or α+ β = 1. (3.5)

Then the imposed restrictions α > 0 (3.1) and β < 1 (2.33) are mutually compatible and(3.3), (3.4) can be rewritten as

at + ωγ〈(u · ∇)b〉 = 0 where γ ≡ 2β − 1 = 1− 2α (3.6)

ω(bτ + (u · ∇)a

)+ bt + ωβ(u · ∇)b = 0 (3.7)

Admixture and Drift in Oscillating Fluid Flows 9

The second term in eqn.(3.6) represents the changes of a due to the averaged nonlineareffects of oscillations. Taking the different values of γ we introduce three qualitativelydifferent asymptotic families of solutions:

γ > 0 (1/2 < β < 1, 0 < α < 1/2) −super-critical oscillations (3.8)

γ = 0 (β = 1/2, α = 1/2) −critical oscillations (3.9)

γ < 0 (−∞ < β < 1/2, 1/2 < α < ∞) −sub-critical oscillations (3.10)

One can notice that super-critical, critical, and sub-critical oscillations have been intro-duced as the asymptotic families of solutions, not as particular solutions (recall that anyparticular solution can be treated as a member of several different asymptotic families).For the family of critical oscillations the averaged oscillatory term in (3.6) has the

same order O(1) as a mean solution. Then the leading terms (for large ω) in (3.6), (3.7)are:

at + 〈(u · ∇) b〉 = 0; (3.11)

bτ + (u · ∇) a = 0. (3.12)

This system gives a single equation for a

at − 〈(u · ∇)(ξ · ∇)〉 a = 0, ξ ≡ uτ (3.13)

which is obtained by T-integration (2.21) of (3.12) and the following substitution of

b(x, t, τ) = −(ξ · ∇) a into (3.11).The critical family of oscillations (3.9) separates the larger families of sub-critical and

super-critical oscillations from each other. For super-critical oscillations γ > 0 and thecorrelation in (3.6) can not be balanced by at = O(1). Then the condition of solvabilityrequires this correlation to vanish, hence the leading order of (3.11), (3.13) is replaced

with 〈(u · ∇) b〉 = 0. It means that the imposed velocity must degenerate (satisfy certainadditional restrictions).Remarks:1. The inspection procedure is aimed to find and classify all possible asymptotic solu-

tions (of the functional form (3.1)) of a given PDE. This procedure also allows to choosea natural small parameter (2.9) with different values of α for different classes of solutions.

2. The critical and super-critical oscillations are most interesting both theoreticallyand practically. We justify this statement with three arguments: (A) Sects.4-7 below arecompletely devoted to these two oscillatory families. The obtained results are mathemat-ically satisfactory and look physically convincing. (B) These families of oscillations arethe least studied, probably due to the reason that for them U → ∞ as ω → ∞ in (2.13);however it is well-known that such an increase of U does not diminish the validity ofasymptotic procedures. (C) In Sect.7.11 we will consider the sub-critical family (3.10)with β = 0 in (2.33),(2.34). We will show that for this family a drift is replaced with apseudo-drift and the general solution is diverging.

4. Family of Critical Oscillations. Purely Oscillatory Velocity.

4.1. Critical asymptotic procedure with α = 1/2

Following (3.9) we accept that (2.33) belongs to a critical asymptotic family:

u(x, t, τ) = u(x, t, τ), δ = ω−1/2, (4.1)

10 V. A. Vladimirov

Then (2.34), (4.1) yield:

Da = ωaτ + at +√ω(u · ∇) a = 0 (4.2)

The small parameter ε1/2 = 1/√ω (2.9) allows to rewrite it as

D2a ≡ aτ + ε(u · ∇) a+ ε2at = 0, D2 ≡ ε2D = D/ω (4.3)

where the subscript in ε1/2 has been dropped. We are looking for the solution of (4.3) inthe form of regular series

a(x, t, τ) =

∞∑

k=0

εkak(x, t, τ), ak ∈ H ∩O(1), k = 0, 1, 2, . . . (4.4)

The substitution of (4.4) into (4.3) produces the equations of successive approximations

a0τ = 0 (4.5)

a1τ = −(u · ∇) a0 (4.6)

anτ = −(u · ∇) an−1 − ∂t an−2, ∂t ≡ ∂/∂t, n = 2, 3, 4, . . . (4.7)

ak = ak(x, t) + ak(x, t, τ), ak ∈ B ∩O(1), ak ∈ T ∩O(1), k = 0, 1, 2, . . .

where (4.6) represents a systematically derived version of (3.12). Let a[n] be a truncatedsolution (4.4) of order n:

a[n](x, t, τ) ≡n∑

k=0

εkak(x, t, τ) (4.8)

Its substitution into (4.3) produces the RHS-residual Res[n]:

D2a[n] = Res[n] (4.9)

4.2. General solution for the first five approximations

The detailed solving of (4.5)-(4.7) for n = 0, 1, 2, 3, 4 is given in Appendix A. The trun-cated general solution a[4] is

a[4] = a0 + εa1 + ε2a2 + ε3a3 + ε4a4 (4.10)

with the tilde-parts ak given by explicit recurrent expressions

a0 ≡ 0, (4.11)

a1 = −(ξ · ∇) a0, (4.12)

a2 = −(ξ · ∇) a1 − (u · ∇) a1τ , (4.13)

a3 = −(ξ · ∇) a2 − (u · ∇) a2τ − aτ1t, (4.14)

a4 = −(ξ · ∇) a3 − (u · ∇) a3τ − aτ2t, (4.15)

ξ ≡ uτ =

∫ τ

0

u(x, t, σ)dσ − 1

2π

∫ 2π

0

(∫ µ

0

u(x, t, σ)dσ

)dµ (4.16)

Admixture and Drift in Oscillating Fluid Flows 11

and the bar-parts ak satisfy the equations(∂t + V 0 · ∇

)a0 = 0 (4.17)

(∂t + V 0 · ∇

)a1 + (V 1 · ∇)a0 = 0 (4.18)

(∂t + V 0 · ∇

)a2 + (V 1 · ∇)a1 + (V 2 · ∇)a0 =

∂

∂xi

(χik

∂a0∂xk

), (4.19)

V 0 ≡ 1

2[u, ξ], V 1 ≡ 1

3〈[[u, ξ], ξ]〉, (4.20)

V 2 ≡ 1

4〈[[V 0, ξ], ξ]〉+

1

2〈[V 0, V

τ

0 ]〉+1

2〈[ξ, ξt]〉+

1

2〈ξdiv u′ + u

′div ξ〉, (4.21)

u′ ≡ ξt − [V 0, ξ], (4.22)

2χik ≡ 〈u′iξk + u′

k ξi〉 = LV 0

〈ξiξk〉, (4.23)

LV 0

f ik ≡(∂t + V 0 · ∇

)f ik − ∂V 0k

∂xmf im − ∂V 0i

∂xmfkm (4.24)

where the operator LV 0

is such that LV 0

f ik = 0 represents the condition for tensorial

field f ik(x, t) to be ‘frozen’ into velocity field V 0(x, t). Three equations (4.17)-(4.19) canbe written as a single advection-pseudo-diffusion equation (with the error O(ε3))

(∂t + V · ∇

)a =

∂

∂xi

(κik

∂a

∂xk

)(4.25)

V = V[2]

= V 0 + εV 1 + ε2V 2, (4.26)

κik = χ[2]ik = ε2χik (4.27)

a = a[2] = a0 + εa1 + ε2a2 (4.28)

where all coefficients are given in (4.20)-(4.24). Eqn. (4.25) shows that the averaged

motion of a represents a drift with velocity V = V[2]

+O(ε3) and pseudo-diffusion withthe matrix coefficient κik = ε2χik +O(ε3).Definition: The term pseudo-diffusion (PD) means that: (i) the evolution of a is de-

scribed by an advection-diffusion-type equation (4.25); (ii) the diffusion-type term ap-pears in (4.25) not in the leading approximation (see (4.19), where it represents a knownsource-type-term in the second approximation); and (ii) the equation (4.25) is valid onlyfor regular asymptotic expansions (4.26)-(4.28).In a[4] (4.10) the expressions for a0, a1, a2, a1, a2, a3, and a4 are given by (4.11)-(4.24),

while a3 and a4 are to be found from higher approximations. The substitution of a[4] into(4.3) produces the RHS-residual of order ε5

D2a[4] = Res[4] = O(ε5) = O(1/ω5/2) (4.29)

One can notice that the residual of orderO(1/ω5/2) forD2 leads to the residualO(1/ω3/2)for the original operator D(4.2).Remarks:1. The statement that (4.11)-(4.24) describe the solution of the problem with arbitrary

initial data is based on the fact that initial data for (4.17)-(4.19) can be chosen arbitrarily.2. We have explicitly solved the equations of the first five approximations. Such persis-

tence does not occur often in the use of asymptotic methods. However, the well-knownexamples are the surface gravity wave, see Stokes (1847), Stoker (1957), Debnath (1994)and the low-Re-solutions for a sphere, see Chester, Breach, and Proudmen (1969). Ourmotivation for doing the cumbersome calculations of Appendix A is based on two our

12 V. A. Vladimirov

results: (i) the first and the second approximations of the critical asymptotic solutionsappear as the main-order terms in super-critical solutions (see Sect.5 ); and (ii) PD (whichqualitatively complements a drift) appears only in the second approximation.

3. The term PD is different from super-diffusion, which means the spreading of an ad-mixture faster than

√t (see Avallaneda & Majda (1992), Volpert, Nec, and Nepomnyashchy (2010)).

5. Super-critical Oscillations

5.1. Super-critical asymptotic procedure with α = 1/3

For critical oscillations (4.1) the drift velocity (4.26) is V ∼ V 0 = O(1) (4.20). From(4.26) one can expect that for the degenerated u such that

V 0 =1

2〈[u, ξ]〉 ≡ 0 (5.1)

the drift is V ∼ εV 1 = O(1/√ω). However, one can find that the drift velocity is

still O(1) in a different asymptotic family. To show it, we accept (5.1) and consider asuper-critical family of purely oscillating velocities

u(x, t, τ) = u(x, t, τ), δ = ω−1/3 (5.2)

where u(x, t, τ) ∈ T ∩O(1) is a given function. Then eqns. (2.34), (5.2) yield:

Da = ωaτ + at + ω2/3(u · ∇) a = 0 (5.3)

A small parameter ε1/3 = ω−1/3 allows to rewrite (5.3) as

D3a ≡ aτ + ε(u · ∇) a+ ε3 at = 0, D3 ≡ ε3D = D/ω (5.4)

where the subscript in ε1/3 is dropped. We are looking for a solution of (5.4) in the formof regular series (4.4) with the redefined ε. The substitution of (4.4) into (5.4) producesthe equations for successive approximations (cf. with (4.5)-(4.7))

a0τ = 0 (5.5)

a1τ = −(u · ∇) a0 (5.6)

a2τ = −(u · ∇) a1 (5.7)

anτ = −(u · ∇) an−1 − ∂t an−3, n = 3, 4, 5, . . . (5.8)

These equations can be solved in the most general form (see Appendix B). Here wepresent the truncated solution a[4] (4.10). For the tilde-part of each approximation onecan obtain explicit recurrent formulae

a0 ≡ 0, (5.9)

a1 = −(ξ · ∇) a0, (5.10)

a2 = −(ξ · ∇) a1 − (u · ∇) a1τ (5.11)

a3 = −(ξ · ∇) a2 − (u · ∇) a2τ (5.12)

a4 = −(ξ · ∇) a3 − (u · ∇) a3τ − aτ1t, ∀a3 (5.13)

Admixture and Drift in Oscillating Fluid Flows 13

and the transport equation (which is valid with the error O(ε3)) for averaged motion(∂t + V · ∇

)a = 0, (5.14)

V = V[2]

= V 1 + εV 2, (5.15)

a = a[2] = a0 + εa1, ∀a2, a3, a4 (5.16)

V 1 = −1

3〈[K, ξ]〉, V 2 = −1

8(κ+K′′), (5.17)

K ≡ [ξ, u], κ ≡ [Kτ, K], K

′′

≡ [[K, ξ], ξ] (5.18)

Eqn.(5.14) means that the averaged motion of a passive admixture (up to the first two

approximations) represents a pure drift with velocity V[2]

(5.15).Hence, in a[4] (4.10) functions a0, a1, a1, a2, a3, and a4 are given by (5.14)-(5.17)

and (5.10)-(5.13), while a2, a3 and a4 are to be found from further approximations. Thesubstitution of a[4] into (5.4) produces the RHS-residual of order ε5

D3a[4] = Res[4] = O(ε5) = O(1/ω5/3) (5.19)

The residual of order O(1/ω5/3) for the operator D3 means that the residual for originaloperators (2.1), (2.4), (2.34), and (5.3) is O(1/ω2/3).

5.2. Super-critical asymptotic procedure with α = 1/4

Next we consider the oscillating velocity fields with first two terms of the drift velocity(4.26) vanishing

V 0 =1

2〈[u, ξ]〉 ≡ 0 and V 1 =

1

3〈[[u, ξ], ξ]〉 ≡ 0 (5.20)

Here one can find that the drift of magnitude O(1) appears within the asymptotic familywith α = 1/4. Let us accept (5.20) and introduce a family of super-critical oscillations

u(x, t, τ) = u(x, t, τ), δ = ω−1/4 (5.21)

where u(x, t, τ) ∈ T ∩O(1) is a given function. Eqns. (2.34),(5.21) give:

Da = ωaτ + at + ω3/4(u · ∇) a = 0 (5.22)

The small parameter ε1/4 = ω−1/4 allows us to rewrite it as

D4a ≡ aτ + ε(u · ∇) a+ ε4 at = 0, D4 ≡ ε4D = D/ω (5.23)

where the subscript in ε1/4 is dropped. We are looking for a solution of (5.23) in the formof regular series (4.4) with the redefined ε. The substitution of (4.4) into (5.23) producesthe equations for successive approximations (cf. with (4.5)-(4.7) and (5.5)-(5.8))

a0τ = 0 (5.24)

a1τ = −(u · ∇) a0 (5.25)

a2τ = −(u · ∇) a1 (5.26)

a3τ = −(u · ∇) a2 (5.27)

anτ = −(u · ∇) an−1 − ∂t an−4, n = 4, 5, . . . (5.28)

Again, these equations can be solved in the most general form (see Appendix B). Herewe present the truncated solution a[4] (4.10). For the tilde-part of each approximation

14 V. A. Vladimirov

one can obtain explicit recurrent formulae

a0 ≡ 0, (5.29)

a1 = −(ξ · ∇) a0, (5.30)

a2 = −(ξ · ∇) a1 − (u · ∇) a1τ (5.31)

a3 = −(ξ · ∇) a2 − (u · ∇) a2τ (5.32)

a4 = −(ξ · ∇) a3 − (u · ∇) a3τ , ∀a3 (5.33)

and the transport equation for the averaged motion(∂t + V · ∇

)a = 0, (5.34)

V = V[2]

= V 2, a = a[2] = a0;

with the same notations as in (5.17), (5.18). Equation (5.34) means that averaged motionrepresents a pure drift with velocity V 2.Hence, a[4] can be written as (4.10) where a0, a1, a0, a2, a3, and a4 are given by

(5.29)-(5.34), while a1, a2, a3 and a4 are to be found from further approximations. Thesubstitution of a[4] into (5.23) produces the RHS-residual of order ε5

D4a[4] = Res[4] = O(ε5) = O(1/ω5/4) (5.35)

The residual of order O(1/ω5/4) for the operator D4 means that the residual for originaloperators (2.1), (2.4), (2.34), and (5.22) is O(1/ω1/4).Remark: One can see that the results for both the super-critical families are simpler

than for the critical one (Sect.4 ) and they are build up from the same elements: if theconditions of degenerations (5.1) or (5.20) are valid, then either V 1 or V 2 (4.26) becomesthe leading O(1)-term in (5.16) or (5.35). At the same time pseudo-diffusion does notappear (within the considered precision).

6. Polynomial Velocity for Critical and Super-Critical Oscillations.

Expressions (4.1), (5.2), (5.21) for velocity have a special form. One can expect thatthe adding of higher-order terms to these expressions may potentially change both thedrift and pseudo-diffusion.

6.1. Critical asymptotic procedure with α = 1/2

Let us consider a critical asymptotic family with a velocity in (2.2),(2.10),(2.34) given as(cf. with (4.1))

u = v +1√ωw +

1

ωr, δ = ω−1/2, (6.1)

where v, w, r ∈ H ∩ O(1) are given functions. This formula might be seen as the firstthree terms of an infinite series, truncated by the omitting of terms O(1/ω3/2). Eqns.(2.34), (6.1) yield:

Da = ωaτ +√ω(v · ∇) a+ (∂t + w · ∇)a+

1√ω(r · ∇) a = 0 (6.2)

The small parameter ε1/2 = 1/√ω allows to rewrite it as

D2a ≡ aτ + ε(v · ∇) a+ ε2(∂t + w · ∇)a+ ε3(r · ∇) a = 0 (6.3)

Admixture and Drift in Oscillating Fluid Flows 15

where D2 ≡ ε2D = D/ω, and the subscript in ε1/2 is dropped. We are looking for thesolution of (6.3) in the form of regular series (4.4). The substitution of (4.4) into (6.3)produces the equations for successive approximations

a0τ = 0 (6.4)

a1τ = −(v · ∇) a0 (6.5)

a2τ = −(v · ∇)a1 − (∂t + w · ∇)a0 (6.6)

a3τ = −(v · ∇)a2 − (∂t + w · ∇)a1 − (r · ∇)a0 (6.7)

The solving of (6.4)-(6.7) follows the same steps as in Appendix A. An important newelement is: the bar-part of eqn. (6.5) is v · ∇a0 = 0. It requires v ≡ 0 as a solvabilitycondition. Therefore the leading term in the prescribed velocity (6.1) must always bepurely oscillatory.Here we present only the truncated solution a[3] (4.10). For the tilde-parts ak with

k = 0, 1, 2, 3 the equations (6.4)-(6.7) yield the recurrent expressions

a0 ≡ 0, (6.8)

a1 = −(ξ · ∇) a0, (6.9)

a2 = −(v · ∇) a1τ − (ξ · ∇) a1 − (η · ∇) a0 (6.10)

a3 = −(v · ∇) a2τ − (w · ∇) a1τ − (6.11)

−(ξ · ∇) a2 − (η · ∇) a1 − (ζ · ∇) a0 − (∂t +w · ∇)aτ1

ξ ≡ vτ , η ≡ w

τ , ζ ≡ rτ , (6.12)

and for the bar-parts ak we obtain the transport equation(∂t + (w + V 0) · ∇

)a0 = 0 (6.13)

(∂t + (w + V 0) · ∇

)a1 + ((r + V 1 + V 12) · ∇)a0 = 0 (6.14)

V 0 ≡ 1

2〈[v, ξ]〉, V 1 ≡ 1

3〈[[v, ξ], ξ]〉, V 12 ≡ 〈[v, η]〉, (6.15)

which can be rewritten (with the error O(ε2)) as a single advection equation(∂t + V · ∇

)a = 0 (6.16)

V = (w + V 0) + ε(r + V 1 + V 12), a = a0 + εa1 (6.17)

Notice that the solution given by (6.8)-(6.17) satisfies eqn. (6.3) with Res[3] = O(ω−2).Remark: In (6.4)-(6.17) we present only four steps (k = 0, 1, 2, 3) of successive approx-

imations since that is enough to give us an instructive correction V 12 to the previouslycalculated drift (4.18), (4.20). The fifth step (k = 4) has been also carried out but forbrevity it is not shown here: it produces a combination of a drift term V 2 (which ismore cumbersome than (4.21)) and pseudo-diffusion with the same coefficients (4.23).The related calculations are similar to those of Appendix A.

6.2. Super-critical asymptotic procedure with α = 1/3

Next, we consider a super-critical asymptotic family with a velocity similar to (6.1)

u = v + ω−1/3w + ω−2/3 r, δ = ω−1/3 (6.18)

where v, w, r ∈ H∩O(1) are given functions. Expression (6.18) might be seen as the firstthree terms in an infinite series, truncated by omitting the terms O(1/ω). Eqns. (2.34),

16 V. A. Vladimirov

(6.18) yield:

Da = ωaτ + ω2/3(v · ∇) a+ ω1/3(w · ∇) a+ (∂t + r · ∇)a = 0 (6.19)

The small parameter ε1/3 = ω−1/3 (2.9) allows to rewrite (6.19) as

D3a ≡ aτ + ε(v · ∇) a+ ε2(w · ∇) a+ ε3(∂t + r · ∇)a = 0 (6.20)

where D3 ≡ ε3D = D/ω and the subscript in ε1/3 is dropped. We are looking for thesolution of (6.20) in the form of regular series (4.4). The substitution of (4.4) into (6.20)produces the equations for successive approximations

a0τ = 0 (6.21)

a1τ = −(v · ∇) a0 (6.22)

a2τ = −(v · ∇)a1 − (w · ∇)a0 (6.23)

a3τ = −(v · ∇)a2 − (w · ∇)a1 − (∂t + r · ∇)a0 (6.24)

The solving of (6.21)-(6.24) follows the same steps as in Appendix B. There are twosolvability conditions

v ≡ 0, V 0 = −w (6.25)

which are required for the existence of the bar-part solutions in (6.22) and (6.23). Onemight see the consideration of Sect.5 as being physically incomplete. Indeed, one canexpect that the results for small V 0 6= 0 are close to that for V 0 = 0. This expectationhas been met and fulfilled by a more general degeneration condition V 0 = −w (6.25)which leads to a more general (than (5.15)) expression for a drift.For tilde-parts ak one can obtain the explicit recurrent expressions which coincide with

(6.9)-(6.11), while the formula (6.12) for a3 should be replaced by

a3 = −(v · ∇) a2τ − (w · ∇) a1τ − (6.26)

−(ξ · ∇) a2 − (η · ∇) a1 − (ζ · ∇) a0 − (w · ∇)aτ1

For the bar-parts one can derive the equation

(∂t + V · ∇)a0 = 0, V = r + V 1 + V 12 (6.27)

which is valid with the error O(ε2) and contains the same V 1 and V 12 as in (6.15).Notice that this solution satisfies the equation (6.20) with Res[3] = O(ω−4/3).Remarks:1. One might additionally consider a polynomial version of velocity for ω1/4-family

(5.21). The related results could be predicted by the analogy between the material ofSect.6.2 and Sect.5.2. For the critical family (6.1) slow motion represents the drift (6.17) of magnitude

O(1) combined with the same pseudo-diffusion of order O(1/ω) as in (4.28), hence theresults are very similar to those of Sect.4.3. All the results of Sects.2-6 can be generalized to an arbitrary flow domain Ω with

fixed boundary ∂Ω. Then (2.26) yields that

V 0 · n = V 1 · n = V 2 · n = u′ · n = 0 on ∂Ω (6.28)

provided u satisfies the no-leak condition. One can also see that if a fluid is incompressiblethen (2.25) yields that all drift velocities in (4.17)-(4.24) are also incompressible:

div u = 0 ⇒ divV 0 = divV 1 = divV 2 = 0 (6.29)

The explicit expressions (4.17)-(4.22) show that due to incompressibility only the last

Admixture and Drift in Oscillating Fluid Flows 17

term in V 2 (4.21) vanishes. Similar (for (6.13)-(6.15)) the incompressibility yields

div v = div w = div r = 0 ⇒ divV 0 = divV 1 = divV 12 = 0 (6.30)

4. For the super-critical family (6.18) with the degeneration (6.25) slow motion is a puredrift with the velocity (6.27) which is ‘upgraded’ to O(1) from O(ε)-term in (6.17). Thequalitatively new addition to a drift (in comparison with a critical family) is a cross-termV 12 ≡ 〈[v, η]〉. It represents an averaged commutator between two mutually independentfunctions v and η; hence V 12 expands the class of available analytical expressions for adrift. For example, for an incompressible fluid 〈[v, η]〉 = ∇×〈v× η〉, so one can see that〈v × η〉 represents an arbitrary vector-potential for V 12.

7. Examples.

7.1. Superposition of two modulated oscillatory fields of the same frequency

The velocity field u (4.1) and ξ ≡ uτ (2.21) are

u(x, t, τ) = p(x, t) sin τ + q(x, t) cos τ (7.1)

ξ(x, t, τ) = −p(x, t) cos τ + q(x, t) sin τ (7.2)

with arbitrary B-functions p and q. The straightforward calculations yield

[u, ξ] = [p, q] (7.3)

so the commutator surprisingly is not oscillating. The drift velocities (4.20), (4.21) are

V 0 =1

2〈[u, ξ]〉 = 1

2[p, q], V 1 =

1

3〈[[u, ξ], ξ]〉 ≡ 0, (7.4)

V 2 =1

8

([P ,p] + [Q, q]

)− 1

4([pt,p] + [qt, q]) + (7.5)

+1

4

(p divP

′+ qdivQ

′+ P

′div p+Q

′div q

),

P ≡ [V 0,p], Q ≡ [V 0, q], P′ ≡ pt − P , Q

′ ≡ qt −Q,

〈ξiξk〉 =1

2(pipk + qiqk) (7.6)

A pseudo-diffusion matrix κik, which follows after the substitution of (7.6) into (4.23).

Remarks:

1. The degeneration V 0 ≡ 0 in an incompressible fluid (6.29),(6.30) corresponds top×q = ∇ϕ with an arbitrary potential ϕ(x, t). For any such fields all results of Sect.5.1are valid.

2. The velocity u (7.1) is general enough to produce any given function V 0(x, t). Toobtain p(x, t) and q(x, t) one has to solve a bilinear first-order PDE

[p, q] = (q · ∇)p− (p · ∇)q = 2V 0(x, t) (7.7)

which represents an underdetermined bi-linear PDE-problem for two unknown functions.

18 V. A. Vladimirov

7.2. Fourier series of modulated fields

The previous example can be generalized by the consideration of a velocity

u(x, t, τ) =∞∑

k=1

pk(x, t) sin kτ + qk(x, t) cos kτ (7.8)

ξ =

∞∑

k=1

−pk

kcos kτ +

qk

ksin kτ (7.9)

The calculations of V 0 lead to an infinite sum of commutators

V 0 =1

2〈[u, ξ]〉 =

∞∑

k=1

1

2k[pk, qk] (7.10)

In particular, (7.10) gives an infinite number of fields u with V 0 ≡ 0. For example, ifpk ≡ 0, ∀k then

V 1 =1

3〈[[u, ξ], ξ]〉 = 1

12

∑

m±n±l=0

1

ml[[qn, qm], ql] (7.11)

where the sum is taken over all sets of three positive integersm,n, l such thatm±n±l = 0.It is clear that for many combinations V 0 ≡ 0 and V 1 6= 0 (see Sect.5.1 ).

7.3. The Stokes drift

This most celebrated example of a drift we consider in some details. We start with thescaling (2.7), (2.8) and the limit (2.14). The dimensional solution for a plane potentialtravelling gravity wave (see Stokes (1847), Lamb (1932), Debnath (1994)) is

u∗ = U u, U =

k∗g∗h∗

ω∗, u = exp(k∗y∗)

(cos(k∗x∗ − τ)sin(k∗x∗ − τ)

)(7.12)

where ω∗, k∗, h∗, and g∗ are dimensional frequency, wavenumber, spatial amplitude, andgravity; (x∗, y∗) are Cartesian coordinates, and τ ≡ ω∗t∗. One can immediately noticethat: (i) the characteristic length is L = 1/k∗; and (ii) the scale U is apparent. Hence thedimensionless scaling parameters (2.8), (2.12) and the asymptotic limot (2.14) appear ask = 1 and

ε1 = 1/Tω∗ = 1/ω → 0, δ ≡ U/ω∗L = g∗h∗/L2ω∗2 = gh/ω2 → 0 as ω → ∞ (7.13)

where one can choose the scale T = T (ω∗) in an arbitrary way, its only mission is toprovide ε1 → 0 as ω → ∞ (see Sect.2.3 ). The simplest way to provide δ = 1/

√ω (4.1) is

to choose an asymptotic family as

T = const, L = const, gh = O(ω3/2) (7.14)

The dimensionless velocity field (7.12) and ξ are

u = Aeky(

cos(kx− τ)sin(kx− τ)

), ξ = Aeky

(− sin(kx− τ)cos(kx− τ)

)(7.15)

where in the chosen system of units A = 1 and k = 1; however, we keep both A and k inthe formulae for tracking its physical meaning. The fields p(x, y), q(x, y) (7.1) are

p = Aeky(

sin kx− cos kx

), q = Aeky

(cos kxsinkx

)(7.16)

Admixture and Drift in Oscillating Fluid Flows 19

The calculations (with the use of (7.3), (7.4)) yield

V 0 = kA2e2ky(

10

), V 1 ≡ 0 (7.17)

which gives V 0 proportional to the classical Stokes drift and a zero value for the firstcorrection to it; the explicit formula for V 2 is not given here for brevity. At the sametime

〈ξiξk〉 = Ξ(x, y, t)δik, with Ξ =1

2A2e2ky (7.18)

which can bee seen as ‘locally isotropic’ oscillations. The matrix of pseudo-diffusion (4.23)is

2χik =(

∂t + V 0 · ∇)δik − 2χ eik

Ξ, 2eik ≡ ∂V 0i

∂xk+

∂V 0k

∂xi

Further calculations show that

χik = −χ

(0 11 0

), with χ ≡ 1

4k2A4e3ky

One can see that the eigenvalues χ1 = −χ and χ2 = χ correspond to ordinary diffusionin one direction and anti-diffusion in the perpendicular direction, so here we have amixed case of pseudo-diffusion. The averaged equation (4.25) (with an error O(ε3)) canbe written as

at + (V 0 + ε2V 2)ax = ε2(χyax + χaxy) (7.19)

a = a0 + εa1 + ε2a2

where V 0 and V 2 are the x-components of corresponding velocities (their y-componentsvanish). This equation has an exact solution a = a(y) (with an arbitrary function a(y)),which is not blurred by pseudo-diffusion.Remarks:

1. In order to avoid confusion one should recall that the choice of asymptotic family(7.14) has nothing to do with physical relations between parameters. Any asymptoticfamily represents only a formal one-dimensional parametrization in the functional spaceof all possible functions u

∗ (7.12). The only aim of such a parametrization is to builda valid asymptotic procedure. Therefore any particular wave (7.12) with given valuesk∗ = k∗0 and ω∗ = ω∗

0 can be considered as a point on an asymptotic curve with k = 1and with variable ω irrespectively to the presence of any dispersion relation between k∗0and ω∗

0 .2. A parameter gh → ∞ as ω → ∞ (7.14) which is a common property for many

asymptotic procedures.3. More general (than (7.14)) asymptotic families are considered in Sect.2.3 and in

(8.65),(8.66).4. Both fields (7.15) are unbounded as y → ∞, but it is not essential for our purposes.

7.4. Spherical ‘acoustic’ wave

A velocity potential for an outgoing spherical wave is

φ =A

rsin(kr − τ) (7.20)

20 V. A. Vladimirov

where A, k, and r are an amplitude, a wavenumber, and a radius in a spherical coordinatesystem. The velocity is purely radial has a form (7.1)

u = p sin τ + q cos τ, (7.21)

p = A

(1

r2cos kr +

k

rsinkr

), q = A

(− 1

r2sin kr +

k

rcos kr

)(7.22)

where u, p, and q are radial components of corresponding vector-fields. The fields ξ and[u, ξ] are also purely radial; the radial component for the commutator is

ξur − uξr = A2k3/r2 (7.23)

where ξ is radial component of ξ and subscript r stands for radial derivative. The drift(7.3),(7.4) is purely radial with

V 0 =A2k3

2 r2, V 1 = 0, V 2 =

A4k5

16r4

(3k2 − 5

r2

)(7.24)

It is interesting that V 0 formally coincides with the velocity caused by a point sourcein an incompressible fluid and for small r the convergence of V is endangered, since V 2

dominates over V 0. Further calculations yield

〈ξ2〉 = A2

2r2(k2 + 1/r2), χ = A4k5/4r2 > 0 (7.25)

where χ stands for the only nonzero rr-component of χik. One can see that in this casepseudo-diffusion appears as ordinary diffusion.

7.5. The V 1-drift.

If V 0 ≡ 0 then the drift of order O(1) is given by (5.15). Let the velocity field (5.2) bea superposition of two standing waves of frequencies ω and 2ω:

u(x, t, τ) = p(x, t) sin τ + q(x, t) sin 2τ (7.26)

ξ(x, t, τ) = −p(x, t) cos τ − 1

2q(x, t) cos 2τ (7.27)

[u, ξ] =1

2[p, q](2 cos τ sin 2τ − cos 2τ sin τ) (7.28)

Hence (4.20) yields

V 0 =1

2〈[u, ξ]〉 ≡ 0, V 1 =

1

3〈[[u, ξ], ξ]〉 = 1

8[[p, q],p] (7.29)

These expressions produce infinitely many examples of flows with O(1)-drift in a super-critical asymptotic family (5.15).

7.6. A plane travelling wave of a general shape

The velocity field (7.1) for a plane wave in three-dimensional space is

u = Af ′(k · x− τ) (7.30)

where A and k are two constant vectors, f is a T-function of a scalar variable, itsτ -periodicity automatically leads to periodicity w.r.t. k · x, primes stand for ordinaryderivatives. Calculations yield

V 0 = A(A · k)〈f ′2〉, V 1 = A(A · k)2〈f ′3〉 (7.31)

Admixture and Drift in Oscillating Fluid Flows 21

It shows that the only case of a zero drift corresponds to a transversal wave A⊥k (anincompressible fluid) and the maximal drift takes place for a longitudinal wave A‖k. Apseudo-diffusion matrix is

χik ≡ 0 for 〈ξiξk〉 = AiAk〈f2〉, (7.32)

Next step is to consider the superposition of two travelling plane waves (7.30):

u = Af ′(k · x− τ) +Bg′(l · x− τ) = −Afτ (k · x− τ)−Bgτ (l · x− τ)

ξ = −Afτ (k · x− τ)−Bgτ (l · x− τ) (7.33)

with constant vectors A, B, k, l. Calculations yield

V 0 = (A · k)A〈f ′2〉+ (B · k)B〈g′2〉+ ((A · l)B + (B · k)A) 〈f ′g′〉 (7.34)

which exhibits the interference (third) term. If f ′ = sin(k · x− τ) and g′ = sin(l · x− τ)then (7.34) gives

2V 0 = (A · k)A+ (B · k)B + ((A · l)B + (B · k)A) cos(k − l)x (7.35)

while V 1 is too cumbersome to be presented here. Also

2〈ξiξk〉 = AiAk +BiBk + (AiBk +AkBi) cos(k − l)x

For two mutually perpendicular longitudinal waves (A‖k)⊥(B‖l) one can obtain

χik = −1

2(AiBk +AkBi)

[(A · k)2 − (B · l)2

]sin(k − l)x (7.36)

which for the plane flow A = (A, 0),B = (0, B) gives

χik = −1

8AB

[(A · k)2 − (B · l)2

](0 11 0

)sin(k − l)x (7.37)

Here we have the eigenvalues χ1 = −χ2 of opposite signs, they are also oscillating in space.Once again (as in Sect.7.3 ), we deal with the sign-indefinite case of pseudo-diffusion. Thisexample can be linked to plane sound waves; the drift (7.31) can give an addition to anacoustic streaming (see e.g. Lighthill (1978b)).

7.7. The drift for the polynomial velocity (6.18)

The expression (6.27) for the O(1)-drift within a super-critical family contains the termgiven in (6.15)

V 12 ≡ 〈[v, η]〉 (7.38)

which is built by two mutually independent oscillating functions. Such a functional free-dom allows to construct broad functional classes of drifts. However, in this case one shouldkeep in mind that the degeneration condition V 0 = −w (6.25) imposes a restriction onthe coefficients of (6.18). For example, one can chose v such that V 0 = −w = 0. In thiscase the function v in (7.38) is not arbitrary. Alternatively, one can consider V 0 = −w

as the definition of w. In this case the functions v and η in (7.38) are indeed mutuallyindependent.

7.8. The Bjorknes configuration of two pulsating point sources

This example is aimed to clarify the global structure of drift motion in an oscillatingflow related to Cook (1882), Hicks (1879), Hicks (1890) and to show that interesting

22 V. A. Vladimirov

oscillating flow, which are different from waves, do exist. An incompressible velocityfrom the class of flows (7.1) in Cartesian coordinates x is given by

q(x) = ∇ 1

|x| , p(x) = ∇ 1

|y| , y = x− l, l = const (7.39)

which represents a superposition of two oscillating point sources. Calculations yield arotationally symmetric w.r.t. the l-axis field:

V 0 =2(x · y)

(|x|2|y|2)2 (|y|2x− |x|2y), V 0 · (x× y) = 0 (7.40)

Let us introduce cylindrical coordinates (ρ, φ, z) with an origin at x = l/2. The relatedODEs (8.3) are

ρ = Mρz, φ = 0, z = −M

2

(l2

4− z2 + ρ2

)(7.41)

l ≡ |l|, M ≡ −36ρ2 + z2 − l2/4

|x|2|y|2|x|2 = ρ2 + (z − l/2)2, |y|2 = ρ2 + (z + l/2)2

The qualitative dynamics of particles described by this system is: (i) two singularitiesat the points x = ±l/2; (ii) any point of the sphere ρ2 + z2 = l2/4 represents anequilibrium; and (iii) the directions of particle motions inside and outside of this sphereare topologically opposite to each other.

7.9. The complexity of particle dynamics for V 0

Now we illustrate the complexity of particle dynamics (8.3) for the zero-order drift ve-locity V 0 (4.20) (see Aref (1984), Ottino (1989), Samelson & Wiggins (2006)). Let anincompressible velocity (7.1) be

p =

cos y0

sin y

, q =

a sin zb sinx+ a cos z

b cosx

(7.42)

where (x, y, z) are Cartesian coordinates, a, b are constants. Either of this fields, takenseparately, produces simple integrable dynamics of particles. Calculations yield

V 0 =

−a sin y sinx− 2b sin y cos zb sin z sin y − a cosx cos yb cos z cos y + 2a sinx cos y

, (7.43)

Straightforward computations for this steady flow exhibit chaotic dynamics of particles.In particular, positive Lyapunov exponents have been observed. This example shows,that the drift created by simple oscillatory field can produce complex dynamics. Sincethe averaged dynamics is chaotic then related results by Arnold (1964), Aref (1984),Ottino (1989), Samelson & Wiggins (2006), Chierchia & Gallavotti (1994) can be ap-plied to it. This example brings up new questions: (i) what is the relationship betweenchaotic motions for the original dynamical system and the averaged one? (ii) can the os-cillatory part of a solution also cause chaotic dynamics? (iii) can a chaotic drift and thepseudo-diffusion of Sect.4 complement each other? (iv) how a chaotic drift can be used inthe theory of mixing? This example has been constructed and computed by A.B.Morgulis(private communications) for the use in this paper.

Admixture and Drift in Oscillating Fluid Flows 23

7.10. The appearance of pseudo-diffusion (PD) due to slow-time-modulations

Let us consider a simple example that clarifies the meaning of pseudo-diffusion.An infinite fluid oscillates as a rigid body with the small displacement of any material

particle x = x(t, τ). The Eulerian coordinate of a particle is

x = x+ x, 〈x〉 ≡ 0, x = εx1, x1 ∈ O(1) (7.44)

where 〈·〉 is the τ -average (2.17); the small parameter ε and the function x1(t, τ) canbe either chosen by us or taken from (8.14)-(8.16). A drift for a rigid-body oscillationsis absent V ≡ 0, hence the mean coordinate x of a particle is not changing with time.For simplicity we also accept that the mean coordinate x coincides with the initial (La-grangian) coordinate X of a particle, hence x(0, 0) = 0.A distribution of a Lagrangian tracer a is given as a = A(X). Then

a(x, t, τ) = A(x− εx1) (7.45)

One can expand both sides of (7.45) as

a0 + εa1 + ε2a2 + . . . = A(x)− εx1i∂A(x)

∂xi+

ε2

2x1ix1k

∂2A(x)

∂xi∂xk+ . . . (7.46)

where an = an(x, t, τ) = an(x, t) + an(x, t, τ). The average 〈·〉x of this equation yields

a0 + εa1 + ε2a2 + . . . = A(x) +ε2

2〈ξiξk〉

∂2A(x)

∂xi∂xk+ . . . (7.47)

where we have changed x1(t, τ) to ξ(t, τ) (4.16), which is valid for the given precision.The t-differentiation of (7.47) gives

a0t + εa1t + ε2a2t + . . . =ε2

2〈ξiξk〉t

∂2a0(x)

∂xi∂xk+ . . . (7.48)

where we have used A = a0 (which follows from (7.47)). One can see that eqns. (4.17)-(4.19) taken for zero drift V = V 0 + εV 1 + ε2V 2 + . . . ≡ 0 coincide with (7.48).It means that in this particular case PD represents a diffusion-like term that appears

in the averaged equations as a correction a2 caused by the curvature of the functiona0 = a0(x). In order to make this explanation clear one can imagine a one-dimensionalcase of (7.44) with A = A(Z) in (7.45) for a single Lagrangian coordinate Z. Then theaveraging of oscillations of the entire graph A = A(Z) in Z-direction evidently produces asecond-order correction (proportional to the curvature AZZ) to the distribution a0(z) =A(z) which occur in the absence of oscillations. Now one might assume that for generaloscillating flows the nature of pseudo-diffusion is the same:Conjecture: Pseudo-diffusion in (4.19), (4.25) is always caused by the oscillations of

Lagrangian tracer a(X , t, τ) with respect to fixed Eulerian coordinates x.According to this conjecture the presence of a drift produces only a logically natural

change from ∂/∂t (for flows (7.45),(7.48) without a drift) to the ‘material’ derivatives(with the drift velocity V in (4.17)-(4.19)). Hence this conjecture looks reliable, it indi-cates that pseudo-diffusion represents a natural as well as necessary term in the averagedequations.Remark: The effects of pseudo-diffusion and diffusion (or anti-diffusion, or anisotropic

diffusion-anti-diffusion) are described by the same equations, hence they are mathemati-cally equivalent to each other. However, pseudo-diffusion is used only with regular asymp-totic procedures.

24 V. A. Vladimirov

7.11. The sub-critical family of oscillations: procedure with α = 1

Let us consider an example of a sub-critical asymptotic family with α = 1 and β = 0.For this family U = L/T = const in (2.13) which might be seen as an advantage for someapplications. Following (3.10) we accept that (2.33) is:

u(x, t, τ) = u(x, t, τ), δ = ω−1, (7.49)

Then (2.34) yields:

Da = ωaτ + at + (u · ∇) a = 0 (7.50)

The small parameter ε1 = 1/ω (2.9) allows to rewrite it as

D1a ≡ aτ + ε(at + u · ∇) a = 0, D1 ≡ εD = D/ω (7.51)

where the subscript in ε1 has been dropped. We are looking for the solution of (7.51)in the form of regular series (4.4) with redifined ε. The substitution of (4.4) into (7.51)produces the equations of successive approximations

a0τ = 0 (7.52)

anτ = −(∂t + u · ∇) an−1, ∂t ≡ ∂/∂t, n = 1, 2, 3, . . . (7.53)

The solving of (7.52),(7.53) yields

a0 ≡ 0, (7.54)

a1 = −(ξ · ∇) a0, (7.55)

a2 = −(ξ · ∇) a1 − (u · ∇) a1τ − aτ1t, (7.56)

a0t = 0, (7.57)

a1t + (V 0 · ∇)a0 = 0 (7.58)

a2t + (V 0 · ∇)a1 + (V+

1 · ∇)a0 =∂

∂xi

(χsubik

∂a0∂xk

), (7.59)

Vsub

1 = V 1 +1

2〈[ξ, ξt]〉+

1

2〈ξdiv ξ〉t, 2χsub

ik ≡ 〈ξi ξk〉t, (7.60)

where V 0 and V 1 are the same as in (4.20). Equations (7.57)-(7.59) can be written as asingle advection-pseudo-diffusion equation (valid with the error O(ε3))

(∂t + V · ∇

)a =

∂

∂xi

(κsubik

∂a

∂xk

)(7.61)

V = εV 0 + ε2Vsub

1 , κsubik = ε2χsub

ik (7.62)

a = a0 + εa1 + ε2a2 (7.63)

Eqn. (7.61) shows that the averaged motion of a represents a pseudo-drift with velocity V

and pseudo-diffusion with the matrix-coefficient κik. The reason for introducing the termpseudo-drift is the same as we used for pseudo-diffusion in (4.25): the drift-like terms in(7.58),(7.59),(7.61) play a part of sources, known from previous approximations.Remarks to Sect.7.11 :1. A simplified (in comparison with (4.23)) expression for pseudo-diffusivity in (7.59)

complies with the interpretation of pseudo-diffusivity of previous subsection.2. The system (7.57)-(7.60) produces mainly diverging solutions. Indeed, (7.57) yields

a0 = a0(x). Let a0(x) 6= const, then (7.58) gives a linear with t growth of a1 for any t-independent V 0, for example for the Stokes drift (see Sect.7.3 ). If we choose a0 ≡ const,then a similar growth appears for a2 etc. A statement that any sub-critical asymptotic

Admixture and Drift in Oscillating Fluid Flows 25

procedure with at ∈ O(1) produces diverging solutions might be considered as a conjec-ture, but its analysis is beyond the scope of this paper.3. This equations of this subsection do not produce the transport equation (drift) in

any approximation.4. The arguments of items 2 and 3 emphasize the significance of critical and super-

critical asymptotic procedures (of Sects.2-7 ).Remarks to all Sect.7 :1. From the examples of Sects.7.1-7.9 one can see that the expressions for the drift

velocity V (4.20), (4.21), (6.17) can provide arbitrary functional form and magnitude nothigher than O(1). In order to make further progress one should prescribe some particularoscillating flows (2.2).2. These examples also show that pseudo-diffusivity in (4.25),(4.26) can appear in

three qualitatively different cases: (i) all positive eigenvalues χi corresponds to ordinarydiffusion; (ii) all negative χi corresponds to anti-diffusion, and (iii) the mixed signs cor-respond to more complex anisotropic evolution with diffusion in some directions andanti-diffusion in the others. The complexity of the problem (4.25) increases if one con-siders the dependence of χi on x and t. The general theory of related linear PDEs (e.g.(7.19)) has not been developed yet, see Polyanin (2002), Myint-U & Debnath (1987).3. The number of our examples is naturally restricted, therefore Sect.7 illustrates our

studies of Sects.3-6 only partially.

8. Links to Other Theories

8.1. TTAM-solution of characteristic equation

We are not aware about any paper which calculates several approximations for a driftby solving the related ODE by two-timing method. Therefore we present the TTAM -versions of such calculations here. Let us return to the original dimensional equation(2.1), (2.2) with an oscillating velocity

u∗ = u

∗(x∗, t∗, τ) (8.1)

where x∗ is an upgraded notation for original Eulerian coordinates which is introduced

here instead of x∗ (see Sect.2 ) since Eulerian coordinates in Lagrangian description alsorepresent the hat-functions (2.15). The oscillating trajectories

x∗ = x

∗(X∗, t∗, τ) (8.2)

(which represent the characteristics for the hyperbolic equation (2.1), (2.4)) can be foundby solving Cauchy’s problem for an ODE

dx∗

ds∗= u

∗(x∗, t∗, τ), x∗|t=0 = X∗ (8.3)

t∗ = s∗, τ = ω∗s∗;d

ds∗=

∂

∂t∗+ ω∗

∂

∂τ(8.4)

where X∗ is the Lagrangian coordinate of a fluid particle and t∗ = 0 automatically leadsto τ = 0. Using the same procedure as for (2.10) we can obtain the dimensionless formof (8.3)

x = x(t, τ ) :dx

ds= ωδu(x, t, τ); t = s, τ = ωs,

d

ds=

∂

∂t+ ω

∂

∂τ(8.5)

which represents the two-timing form of a dynamical system for the motion of particles,see Arnold (1964), Aref (1984), Ottino (1989), Samelson & Wiggins (2006). We accept

26 V. A. Vladimirov

(2.33) and introduce a test-solution for our inspection procedure

x(t, τ) = x0(t) +1

ωαx1(t, τ), α > 0; δ = ωβ−1, β < 1 (8.6)

that is motivated similarly to (3.1) (if one takes x0 6= 0, then τ ceases to be a fastvariable). The substitution of (8.6) into (8.5) yields

(ω

∂

∂τ+

∂

∂t

)(x0 +

1

ωαx1

)= ωβu

(x0 +

1

ωαx1

)(8.7)

The decomposing of the RHS into Taylor’s series with the retaining of two leading termsgives

x0t + ω1−αx1τ + ω−αx1t = ωβu+ ωβ−α(x1 · ∇)u (8.8)

where u = u(x(t, τ), t, τ). The bar- and tilde- parts of this equation are

x0t + ω−αx1t = ωβ−α〈(x1 · ∇)u〉 (8.9)

ω1−αx1τ + ω−αx1t = ωβu+ ωβ−α(x1 · ∇)u + ωβ−α(x1 · ∇)u (8.10)

where we have used the brace notation (2.28). The leading terms in these equations are

x0t = ωβ−α〈(x1 · ∇)u〉 (8.11)

ω1−αx1τ = ωβu (8.12)

Here the τ -average 〈·〉 does not carry any superscript, since the spatial independentvariable is absent; at the same time u depends on an unknown function x0(t). Theequation (8.12) gives α + β = 1, after that (8.11) brings us back to the same notionsof super-critical, critical, and sub-critical asymptotic families (3.8)-(3.10). For a criticalfamily α = β = 1/2 and ε = ε1/2 = 1/

√ω, which produces an asymptotic problem

(ω

∂

∂τ+

∂

∂t

)x =

√ωu, x = x0 +

1√ωx1 +

1

ωx1 + . . . (8.13)

or

xτ = −ε2xt + εu(x, t, τ), x = x0 + εx1 + ε2x1 + . . . (8.14)

The successive approximations are

x0τ = 0 (8.15)

x1τ = u0 (8.16)

x2τ + x0t = (x1 · ∇)u0 (8.17)

x3τ + x1t = (x2 · ∇)u0 +1

2x1ix1k

∂2u0

∂x0i∂x0k(8.18)

x4τ + x2t = (x3 · ∇)u0 + x1ix2k∂2u0

∂x0i∂x0k+

1

6x1ix1kx1j

∂3u0

∂x0i∂x0k∂x0j(8.19)

Admixture and Drift in Oscillating Fluid Flows 27

where u0 ≡ u(x0, t, τ). The system (8.15)-(8.19) can be solved using the same method as(4.5)-(4.7) and (8.56)-(8.59). Here we only present the solution for the averaged motion

x0t = U0 (8.20)

x1t = (x1 · ∇)U 0 +U1

x2t = (x1 · ∇)U 1 + (x2 · ∇)U0 +U2

U0 = 〈(x1 · ∇)u〉, (8.21)

U1 = −〈(x1 · ∇)(u · ∇)x1〉 =1

3〈[[u, x1], x1]〉 (8.22)

U2 = 〈(x3 · ∇)u〉+⟨x1ix2k

∂2u(x)

∂xi∂xk

⟩− 1

2

⟨x1ix1ku1j

∂3x(x)

∂xi∂xk∂xj

⟩(8.23)

which can be rewritten as

xt = U + (l · ∇)U (8.24)

x = x0 + εx1 + ε2x1 + . . . , l ≡ x− x0

U = U0 + εU1 + ε2U2 + . . .

One can see that the system (8.24) describes two kinds of motion: in one every materialpoint moves with velocity U ; in the other every small material arc δl is stretched by thesame velocity field according to a standard law, see Batchelor (1967).

8.2. TTAM-approach to GLM-kinematics

The most general formula for a drift was obtained inGLM -theory by Andrews & McIntyre (1978),which is referred hereafter as AM. The direct comparison of the results of Sects.2-7 withthose of AM is not feasible due to the following conceptual differences: (i) GLM -theoryin AM is not fully adapted to the two-timing method; (ii) AM uses one small parameter(the amplitude of x) while we operate with two small parameters; (iii) AM employs adifferent averaging operation; and (iv) AM expresses a drift in terms of generalized La-grangian displacements x (8.37), not a given velocity field (as in our case). We develophere the TTAM -version of GLM -kinematics with the aim to compare it with the resultsof Sects.2-7.At the beginning of this subsection we use dimensional variables, but for brevity we

suppress the asterisks; the use of dimensionless variables (starting from (8.48)) will beadditionally notified. The core of the whole GLM -theory is a transformation of (8.3) intoa PDE. In order to perform this transformation, we switch to mutually independent t, τ(see Comment C to (2.16)) and introduce a τ -averaged trajectory (8.2) as

x = x(X, t) = 〈x〉X (8.25)

where 〈·〉X emphasises that the τ -average (2.17) is performed for fixed X. From (8.3) weget x(X , 0) = X. Equations (8.3), (8.25) give us three pairs of main kinematical items:(i) two kinds of trajectories for a selected particle

x = x(X , t, τ) - the family of exact trajectories, (8.26)

x = x(X , t) - the averaged trajectory, (8.27)

with x(X , 0, 0) = x(X, 0) = X, (8.28)

(ii) two kinds of velocities

u ≡ ∂x/∂s|X – the family of exact velocities of a particle, (8.29)

ν ≡ ∂x/∂s|X – the averaged velocity of a particle

28 V. A. Vladimirov

(iii) two Eulerian expressions for the same material derivative

∂/∂s|X = ∂/∂s|x + u · ∇ = ∂/∂s|x + ν · ∇ (8.30)

Relations (8.26)-(8.30) show that for Lagrangian description (X, t) one can use two differ-ent Eulerian descriptions (x, s) and (x, s) where x and ν are called the GLM-coordinateand GLM-velocity. One also should keep in mind that in (8.29),(8.30) according to thechain rule

∂/∂s|X = ∂/∂t|X,τ + ω ∂/∂τ |X,t (8.31)

and similar equalities for fixed x or x. The existence of one-to-one mapping between X

and x represents a key postulate of classical fluid dynamics; it guarantees the existenceof the unique inverse to (8.26) function

x = x(X , t, τ) ⇔ X = X(x, t, τ); 0 < J0 < ∞, J0 ≡ ∂(x1, x2, x3)

∂(X1, X2, X3)(8.32)

with a Jacobian J0. After that the function

x = x(x, t, τ) (8.33)

can be obtained by the completely legal substitution of X = X(x, t, τ) (8.32) intox = x(X , t) (8.25).The additional assumption (specific for GLM ) is an assumption of invertibility of (8.27)

x = x(X, t) ⇒ X = X(x, t); 0 < J1 < ∞, J1 ≡ ∂(x1, x2, x3)

∂(X1, X2, X3)(8.34)

The coordinate x is defined by the averaging operation (8.25), hence it does not representany physical motion; (8.34) represents an assumption which can be valid only under someadditional restrictions. For the GLM -theory this invertibility is absolutely required; letus also accept it and show how it leads to GLM -kinematics.The substitution of the inverse function X = X(x, t) (8.34) into x = x(X, t, τ) (8.26)

produces the key (for the GLM -theory) relation between x and x

x = x(x, t, τ); 0 < J2 < ∞, J2 ≡ ∂(x1, x2, x3)

(x1, x2, x3)(8.35)

that is inverse to (8.33). The functions (8.33), (8.35) give us one-to-one mapping betweentwo systems of Eulerian coordinates x and x; since J2 = J0J

−11 , the conditions for the

Jacobians in (8.34) and (8.35) are equivalent to each other, provided (8.32) is valid.An important property of the averaging operation (8.25)

〈·〉x = 〈·〉X , (8.36)

follows from the fact that τ is not involved into transformations (8.34),(8.25) betweenX into x. In other words, the τ -averaging operation commutes with the changing ofvariables (8.34). Taking 〈·〉x of (8.35) we get

x(x, t, τ) = x+ x(x, t, τ) with 〈x〉x = x, 〈x〉x = 0 (8.37)

where 〈x〉x = x is valid by virtue of (8.25) and (8.36). Simultaneously (8.37) representsthe definition of x which automatically possesses zero τ -average; in AM x is calledgeneralized Lagrangian displacement.Applying two Eulerian forms of material derivative (8.30) to both sides of (8.37) we

obtain

(∂/∂s|x + u · ∇)x = u(x, t, τ) = (∂/∂s|x + ν · ∇)(x+ x(x, t, τ)) (8.38)

Admixture and Drift in Oscillating Fluid Flows 29

which can be rewritten as(∂

∂t+ ω

∂

∂τ+ ν · ∇

)(x+ x(x, t, τ)) = u(x+ x(x, t, τ), t, τ) or (8.39)

(∂

∂t+ ω

∂

∂τ+ ν · ∇

)x(x, t, τ) = u(x+ x(x, t, τ), t, τ) − ν(x, t) (8.40)

where (8.39) represents the equation for characteristics (8.3) written as a PDE in vari-ables x, t, τ . We call either (8.39) or (8.40) the two-timing form of Andrews-McIntyre-Kinematics-Equation(AMKE); it contains unknown functions x(x, t, τ) and ν(x, t). Thedescription of fluid kinematics by (8.39) is rather unusual: AMKE describes an effectivemedium which moves with the averaged velocity ν that includes both an advective ve-locity and a drift. The exact motion of a material particle in this medium is describedby two different fields: the averaged motion is described by GLM -velocity ν(x, t) andoscillatory displacement (from the GLM -position x) is given by x(x, t, τ).The first step in solving (8.40) is straightforward: its bar-part 〈·〉x gives us GLM -

velocity ν expressed in the terms of an original velocity

ν(x, t) = 〈u(x+ x(x, t, τ), t, τ)〉x, (8.41)

which represents the main kinematical formula by AM. The average operation in the RHSof (8.41) is called GLM-averaging. GLM -averaging of a function a(x, t, τ) is denoted byAM as

aL = aL(x, t) ≡ 〈a(x+ x(x, t, τ), t, τ)〉x (8.42)

The advection of a Lagrangian marker is described by the equation (2.1)(∂/∂s|x + u(x, t, τ) · ∇

)a(x, t, τ) = 0 (8.43)

Due to (8.30), we replace a material derivative in (8.43) with ∂/∂t|x + ν · ∇ and use(8.37)

(∂/∂s|x + ν · ∇)a(x+ x(x, t, τ), t, τ) = 0 (8.44)

Then 〈·〉x yields

(∂/∂t|x + ν · ∇)aL = 0 (8.45)

This remarkable equation says that the GLM -average aL (8.42) in the GLM -coordinatesx (8.25), (8.27) is purely advected with the GLM -velocity ν (8.41). The drift by AM is

U(x, t) = ν(x, t)− 〈u(x, t, τ)〉x = 〈u(x+ x(x, t, τ), t, τ)〉x − 〈u(x, t, τ)〉x (8.46)

For a purely oscillating field u = u the average 〈u〉x ≡ 0, hence

U = ν(x, t) = 〈u(x+ x(x, t, τ), t, τ)〉x (8.47)

To follow the route used by AM we can take x of a small amplitude and decompose (8.46)and (8.47) into Taylor’s series; however in the two-timing case this problem contains twosmall parameters and is richer in asymptotic procedures. In order to describe them wewrite the dimensionless form of AMKE (8.39) which is obtained by the same procedureas (2.10)

(∂

∂t+ ω

∂

∂τ+ ν · ∇

)(x+ x(x, t, τ)) = ωδu(x+ x(x, t, τ), t, τ) (8.48)

where the given velocity field is taken as a purely oscillatory one. Starting from (8.48)and further in this section we use only dimensionless variables and functions, keeping the

30 V. A. Vladimirov

same notations without asterisks. Using the same inspection procedure as in Sect.3 weaccept (2.33) and introduce a test-solution (similar to (3.1))

x = x+ x = x+1

ωαy(x, t, τ), α = const > 0 (8.49)

where x ∈ B ∩ O(1), y ∈ T ∩ O(1) and the amplitude of an oscillating part is given bythe small parameter εα = 1/ωα (2.9). Hence we obtain

(∂

∂t+ ω

∂

∂τ+ ν · ∇

)(x+

1

ωαy(x, t, τ)

)= ωβu

(x+

1

ωαy(x, t, τ)

)(8.50)

Decomposing the right-hand side of (8.50) into Taylor’s series and omitting the terms oforder O(1/ω2α) and above, we have

ν + ω1−αyτ +1

ωαDν y = ωβu(x, t, τ) + ωβ−α(y · ∇)u(x, t, τ) (8.51)

The bar-part of this equation is

ν = ωβ−α〈(y · ∇)u〉 (8.52)

while the difference between (8.51) and (8.52) produces the tilde-part of the equation.To make a meaningful equation out of the tilde-part (cf. with (3.4)) it is necessary toaccept that dominating terms are of the same order:

α+ β = 1 and yτ = u(x, t, τ) (8.53)

Equations (8.52) and (8.53) brings us back to the same notions of super-critical, critical,and sub-critical asymptotic families (3.8)-(3.10) where a critical family is characterisedby α = β (a classical AM case corresponds to a sub-critical family with α = 1 andβ = 0). Then AMKE (8.39) for the critical asymptotic family α = β = 1/2 takes form

(ω

∂

∂τ+

∂

∂t+U · ∇

)(x+ x(x, t, τ)) =

√ωu(x+ x(x, t, τ), t, τ) (8.54)

x(x, t, τ) =1√ωx1(x, t, τ) +

1

ωx2(x, t, τ) + . . .

U(x, t) = U0(x, t) +1√ωU1(x, t) +

1

ωU2(x, t) + . . .

The use of ε = ε1/2 = 1/√ω (2.9) transforms (8.54) into

(∂

∂τ+ ε2

∂

∂t+ ε2(U0 + εU1 + ε2U2 + . . . ) · ∇

)(x+ εx1 + ε2x2 + . . .

)=

= εu(x+ εx1 + ε2x2 + . . . , t, τ) (8.55)

The first four approximations of this equation are:

x1τ = u(x, t, τ) (8.56)

x2τ +U0 = (x1 · ∇)u(x, t, τ) (8.57)

x3τ + x1t +U1 + (U0 · ∇)x1 = (x2 · ∇)u(x, t, τ) +1

2x1ix1k

∂2u(x, t, τ)

∂xi∂xk(8.58)

x4τ + x2t +U2 + (U1 · ∇)x1 + (U0 · ∇)x2 = (8.59)

= (x3 · ∇)u(x, t, τ) + x1ix2k∂2u(x, t, τ)

∂xi∂xk+

1

6x1ix1kx1j

∂3u(x, t, τ)

∂xi∂xk∂xj

Admixture and Drift in Oscillating Fluid Flows 31

These equations can be solved in the same way as the equations in Sect.4. The solutionsare

x1 = uτ (8.60)

x2 = (x1 · ∇)uτ (8.61)

x3 = xτ1t + (U0 · ∇)xτ

1 + (x2 · ∇)uτ + 1

2

x1ix1k

∂2u(x)

∂xi∂xk

τ

(8.62)

with the same U0,U1, and U2 as in (8.21)-(8.23).

Remarks and discussion:

1. In (8.48) we are looking for the dimensionless drift of order one ν = O(1) that (aswe know from Sects.4 and 5 ) is true for critical and super-critical oscillations. The moresystematic (but more cumbersome) approach is to replace ν in (8.48) by ωδν (which isrequired by (8.47)) and to derive rigorously that the first non-vanishing term in ωδν isO(1).

2. The super-critical versions of the problems considered in Sects.8.1,8.2 can be solvedsimilarly to the problems in Sects.5,6.

3. The expression (8.20),(8.21) was obtained by Yudovich (2006).

4. The theories of both Sect.8.1 and Sect.8.2 operate with the average taken for afixed Lagrangian coordinate X. Indeed, it is the only possibility for the problem (8.5):this problem describes dynamics of a single particle. As to AMKE (8.39), it explicitlyuses the average for fixed x which is the same as for fixed X, see (8.36).

5. One can observe a strong resemblance between the system (8.15)-(8.19) and thesystem (8.56)-(8.59). In (8.15)-(8.19) terms with U0,U1, . . . in LHS are absent andTaylor’s series in RHS contain x = x+ x (instead of x in (8.56)-(8.59)). These differencesare due to the fact that x is an independent variable in (8.56)-(8.59) but in (8.15)-(8.19)it play a part of an unknown function. However one can show that these two systemsare mathematically equivalent to each other, and the drift velocity U in Sect.8.1 andin Sect.8.2 is the same. These properties are natural, since AMKE (8.39) represents atransformed ODE for characteristics (8.3) (see (8.25)-(8.39)).

6. Two sets of formulae for drift velocities (4.16), (4.20) and (8.60), (8.21),(8.22) lookidentical if one makes a correspondence

u(x, t, τ) ↔ u(x, t, τ), ξ(x, t, τ) ↔ x(x, t, τ), (8.63)

V 0(x, t) ↔ U0(x, t), V 1(x, t) ↔ U1(x, t)

hence one may conclude that our formulae for the zeroth and first approximations ofa drift are the same as AM. However this resemblance is misleading, at least partially.The crucial point is the use of the GLM -coordinates x (8.27) in GLM -kinematics vs. theoriginal Eulerian coordinates x ≡ x (8.26) in Sects.2-7. It leads to the different definitionsof the averaging operations with fixed x vs. fixed x. For example, the expressions x1 = u

τ

(8.60) and ξ ≡ uτ (4.16) look identical, however they are different since the former is

expressed as a function of x and the latter as a function of x ≡ x: these variables areconstant during the T-integrations. The implications of such a difference lay beyondthe scope of this paper. At this stage one can see that: (i) the similarity between theexpressions in (8.63) indeed corresponds to the close results for the zeroth and firstapproximations (due to the smallness of the difference x = x − x); (ii) the expressionsfor U2 and V 2 do not exhibit the similarity shown in (8.63).

7. One can see that, in order to express (8.45) in the spirit of Sects.2-7, aL has to be

32 V. A. Vladimirov

decomposed into Taylor’s series for small x. The resulted equations is

(∂/∂t|x + ν · ∇)R = 0, (8.64)

R ≡ 〈a(x, t, τ)〉x +

⟨xi

∂a(x, t, τ)

∂xi

⟩x

+1

2

⟨xixk

∂2a(x, t, τ)

∂xi∂xk

⟩x

+ . . .

where x(x, t, τ), ν(x, t), and a = a + a must be expressed as in (8.54) and (4.4). Thenthe terms of the type 〈xi∂a/∂xi〉 should be calculated with the use of the full equation(8.44). Such calculations will produce Riemann’s invariant R which is constant along theaveraged characteristic curves (or R is transported with the drift velocity).8. It is apparent that the establishing of one-to-one correspondence between (4.25) and

(8.64) is not feasible. Indeed, (4.25) has a ‘dissipative’ pseudo-diffusion term which cannot be incorporated in the transport equation (8.64). This fact should be expected fromSect.7.10, where we have shown that pseudo-diffusion appears due to oscillations of La-grangian coordinates with respect to fixed Eulerian coordinates. Hence, pseudo-diffusion(which is a product of Eulerian averaging) is not compatible with (8.64) (which is aproduct of Lagrangian averaging, see the item 4 above). This incompatibility complieswith a general understanding that different averaging operations preserve and loose orig-inal information differently. One can conclude that the averaged fields in TTAM -form ofGLM -theory and in the theory of Sects. 2-7 contain complementary (or just additionalto each other) information. It opens the opportunity to look for the advantages given bythe presence of these two descriptions.

8.3. Links to other classical theories

The classical drift velocity. The classical expression for a drift velocity is given by the for-

mulae (26) of Longuet-Higgins (1953) or by (5.13.21) of Batchelor (1967). Our V 0 (4.20)will be proportional to this expression if it is rewritten with the use of τ -periodicityand integration by parts. In fact, the conjugated form 〈(ξ · ∇) v〉 of V 0 = −〈(v · ∇) ξ〉(A 17) coincides with the classical expression. Consequently V 0 gives the correct expres-sion for the drift in a potential travelling wave (Stokes (1847)). At the same time, thereare four main differences between our results and that of the quoted classical papers: (i)we describe all involved variables and fields precisely, within the two-timing framework;(ii) we employ two small parameters instead of one; (iii) we obtain different magnitudesfor a drift while in the classical paper the leading term is quadratic in small amplitude(which is also obtainable within our approach); (iv) we obtain two more formulae fordrift velocities as well as find pseudo-diffusion.The Krylov-Bogoliubov method. The Krylov-Bogoliubov averaging method (KBAM),

see Bogoliubov & Mitropolskii (1961), Krylov & Bogoliubov (1947), Sanders & Verhulst (1985),is aimed to solve problems for special classes of ODEs with oscillating coefficients. Thismethod can be straightforwardly used for the calculating of the averaged equations ofcharacteristic curves (8.5). We have exploited this option: the required calculations arerather cumbersome, therefore we formulate here the results only. The first KBAM -termfor a drift velocity coincides with U0 (8.20), while the obtaining of the next two termsrequires substantially more KBAM analytical calculations than TTAM ones. Here wemake two comments: (1) KBAM as well as TTAM calculations of Sect.8.1 give the av-eraged equations for characteristic curves. Within these two methods applied to an ODEthe problem of finding the averaged Riemann invariants cannot be addressed. (2) Theareas of applicability of TTAM are much broader than KBAM : for example TTAM canoperate with PDEs and incorporate molecular diffusivity to the governing equations (seeSect.9 ).

Admixture and Drift in Oscillating Fluid Flows 33

The homogenization theory. TTAM has the same methodological roots as the methodof homogenization (see Bensoussan, Lions and Papanicolaou (1978), Berdichevsky, Jikov, and Papanikolaou (1999)),which represents a version of the two-scale method. The long-wave ‘homogenization’ solu-tions for the transport of scalar admixture obtained by McLaughlin, Papanicolaou & Pironneau (1985),Vergassola & Avellaneda (1997), Frisch (1995) show the appearance of a ‘turbulent’ dif-fusion matrix, which is always positive-definite. In order to establish further links weconsider the case when the given velocity does not contain high frequency oscillationsand can be expressed as u1(x, t/ω, t) (or just u = u1(x, t)) with the 2π-periodic de-pendence on t. Let us show that in this case the problem can be transformed to theconsidered in Sects.2-7.Problem A: The dimensionless form of the transport equation (2.10), (2.34) for a purely

oscillating velocity(

∂

∂s+ ωβu · ∇

)a = 0,

∂

∂s=

∂

∂t+ ω

∂

∂τ(8.65)

where a = a(x, t, τ), u = u(x, t, τ), β = const, the mutually dependent time-variablesare t = s, τ ≡ ωs.Problem B: A similar equation for different time-scales is

(∂

∂s+ ωβ1u1 · ∇

)a1 = 0,

∂

∂s=

1

ω

(∂

∂t1+ ω

∂

∂τ1

)(8.66)

where a1 = a1(x, t1, τ1), u1 = u(x, t1, τ1), β1 = const, the mutually dependent time-variables are t1 = s/ω, τ1 ≡ s.One can see that the equations (8.65) and (8.66) are mathematically identical to each

other if one introduces the link ‘Problem A↔Problem B ’ as

t ↔ t1, τ ↔ τ1, β ↔ β1 + 1, at, aτ ∈ O(1) ↔ at1 , aτ1 ∈ O(1) (8.67)

After such replacements any solution of (8.65) simultaneously produces a solution of(8.66). Hence this rescaling procedure delivers a counterpart a(x, t1, τ1) of a(x, t, τ): inthe Problem B (8.66) the imposed oscillations have the frequency O(1) and the slow-time-scale O(1/ω). The link (8.67) allows us to add one more solution for each solutionin Sects.2-7. The main motivation behind the rescaling (8.67) is: the only solutions con-sidered in the homogenisation theory, see Bensoussan, Lions and Papanicolaou (1978),Berdichevsky, Jikov, and Papanikolaou (1999) are of the type (8.66).The presence of molecular diffusivity. Let µ∗ and µ be the dimensional and dimension-

less coefficients of molecular diffusion. Eqn. (2.1) describes the advection-diffusion of ascalar admixture in an incompressible fluid after adding the term µ∗∇∗2a to the RHS.All results of Sects.2-6 can be straightforwardly generalized to the flows with µ = O(1).Such generalization is achieved by replacing operators ∂/∂s∗ → ∂/∂s∗ − µ∗∇∗2 in (2.1)and ∂/∂t → ∂/∂t − µ∇2 in (2.4) and in all subsequent formulae The limit µ → 0 is aregular one for our strictly regular asymptotic procedures (4.4), so all the problems withµ 6= 0 have the cases studied in Sects.2-6 with µ ≡ 0 as their limits.Magnetohydrodynamics (MHD). One can also develop the same asymptotic procedures

and to derive the averaged equations similar to (4.17)-(4.22) for a vectorial passive admix-ture such as magnetic field h∗(x∗, s∗) in the kinematic MHD-dynamo problem for a givenoscillating velocity field, see Moffatt (1978). In this case one can start with equations

∂h∗/∂s∗ + [h∗,u∗] = 0, divh∗ = 0, divu∗ = 0 (8.68)

which replace (2.1). Taking the same velocity field (4.1) and repeating the same steps as

34 V. A. Vladimirov

in Sect.4 lead to the averaged equation (valid with the error O(ε3))

ht + (V · ∇)h − (h · ∇)V =∂

∂xi

(κik

∂h

∂xk

), divh = 0 (8.69)

h = h0 + εh1 + ε2h2, V = V 0 + εV 1 + ε2V 2, κik = ǫ2χik

where all coefficients are the same as in (4.17)-(4.22). For this problem we have per-formed the detailed calculations of V 0 and V 1, while the expressions for V 2 and κik

represent reliable conjectures. In these calculations one should essentially use (2.24) and(2.32). Notice that eqn. (8.69) describes evolution of h(x, t) for an arbitrary spatialscale L, while the homogenisation approach to the same problem (see e.g. Frisch (1995),Zheligovsky (2009)) operates with long-wave solutions. However, the rescaling of spatialvariables (similar to one for time variables in (8.65),(8.66)) is applicable.

9. Discussion and Conclusions

1. The motivation behind this paper is to study in full a variety of asymptotic solutionsand procedures for the advection of a scalar or vector field in an oscillating flow. Themain result of the paper is the general understanding of the differences between differentclasses of asymptotic solutions.2. The author is not aware about any research paper entirely devoted to the general

analysis of motions of a scalar (or vector) admixture in high-frequency oscillating flows.At the same time this topic is exploited in many papers in the solutions of particularapplied problems (e.g. Magar & Pedley (2005)). We hope, that our general study canunderpin further applied studies.3. Several well-known papers are devoted to the motion of a scalar admixture in spa-

tially oscillating flows, e.g.McLaughlin, Papanicolaou & Pironneau (1985), Vergassola & Avellaneda (1997),and a review in Frisch (1995). These authors have considered only one asymptotic familyof solutions (following to Bensoussan, Lions and Papanicolaou (1978)). Therefore it canbe useful to study both high-frequency and short-wave problems at a greater level ofgenerality and then to consider different asymptotic procedures for flows that oscillateboth in space and time.4. In our paper we are dealing with the systematic calculations of solutions. A few

restrictions have been accepted at the very beginning, the rest of the paper does not con-tain any physical input. Hence, our results require physical interpretations, explanations,and applications.5. The asymptotic validity of our results is provided by the fact that we build only

the solutions for high frequency ω. In our approach ω is not linked to any dynamicalequations. Instead, we require that dimensional frequency ω∗ is high in comparison withthe inverted slow-time-scale 1/T (2.7) of an arbitrary prescribed velocity field (2.2). Ifvelocity field (2.2) is purely oscillatory, then the choice of T is not unique, which isexplained in (2.37),(2.38).6. All results of this paper have been obtained for the τ -periodic functions from the

class H (2.15), which is closed with respect to all used operations. One can try to con-sider more complex classes of quasi-periodic, non-periodic, or chaotic solutions. Thegeneralization of our results to quasi-periodic solutions (containing several τ -periods)looks rather straightforward, provided one can deal with resonances. At the same time,chaotic oscillations (containing low ω in their spectrum) can be hardly treated by ourmethod since ω enters the denominators of asymptotic series (the apparent requirementfor the applicability of our method is separating the frequency spectrum from zero).

Admixture and Drift in Oscillating Fluid Flows 35

Some generalizations of our results can be achieved after replacing the integration overτ0 < τ < τ0 + 2π by integration over −∞ < τ < +∞ in the definition of average (2.17)(as it has been widely accepted for the spatial average in the homogenization theory, seeBerdichevsky, Jikov, and Papanikolaou (1999)).7. In principle, TTAM allows to produce the approximate asymptotic solutions with

as small an error (RHS-residual) as needed. However, the next logical step is more chal-lenging: one has to prove that a solution with a small RHS-residual is close to the exactone. This problem is equivalent to the presence of a small additional force. Such proofshad been performed by Simonenko (1972), Levenshtam (1996) for vibrational convection.Similar justifications of TTAM for other oscillating flows are not available yet.8. Different asymptotic procedures ε1/2, ε1/3, ε1/4, etc. correspond to different asymp-

totic paths on the plane of two scaling parameters (2.12). Two additional asymptoticpaths represent two successive limits: (i) first δ → 0 and then 1/ω → 0 or (ii) first1/ω → 0 and then δ → 0 (these paths are not considered in this paper but they areworth studying). The relevance of different available paths to particular physical situa-tions represents a major problem for further studies. At this stage we can mention onlythat different asymptotic paths correspond to different relations between physical pa-rameters. For example, δ = ω−1/2 (4.1) and δ = ω−1/3 (5.2) in their dimensional form(2.8) give

U = LT−1/2ω∗1/2 and U = LT−1/3ω∗2/3 (9.1)

correspondingly. For the asymptotic procedures with large ω∗ (2.13) it means that theimposed oscillatory velocity (2.2) is higher in the latter (super-critical) case.9. To be able to compare different flows physically one might introduce a drift-efficiency

of an oscillatory flow as the ratio

Eα ≡ (amplitude of drift velocity)/(amplitude of velocity oscillations)

with α defined in (2.9). The dimensionless drifts in both cases (9.1) are O(1) and indimensional form they are of order L/T . Then one might conclude that the efficienciesof (9.1) are E1/2 = (ω∗T )−1/2 and E1/3 = (ω∗T )−2/3 correspondingly, which indicatesthat the critical family solution (4.1) produces stronger drift than the super-critical one(5.2). However, this comparison is not fair since the solution (5.2) is valid only when theleading term in (4.1) vanishes (5.1). Taking the degeneration into account one obtainsthe efficiencies of (9.1) as

Ed1/2 = (ω∗T )−1 and E1/3 = (ω∗T )−2/3

where Ed1/2 represents the drift-efficiency of the degenerated critical solution (4.1),(5.1).

Now, one can conclude that a super-critical solution is more drift-efficient than a criticalone.10. The appearance of small pseudo-diffusion (PD) in the high-frequency asymptotic

problem (4.2), (4.17)-(4.27) is an accurate and qualitatively new result. One can maketwo conjectures: (i) a solution can slowly self-concentrate (due to small negative pseudo-diffusivity) or it can undergo unusual anisotropic evolution (due to sign-indefinite pseudo-diffusivity); and (ii) the maximum principle can be violated (since the original equation(2.1) expresses the conservation of a in each fluid particle, hence the values of sup a andinf a do not change with time). One can argue that the conjecture (ii) is not valid due tothe explanations given in Sect.7.10, while (i) requires additional studies.11. It is worth introducing the finite molecular diffusivity µ = O(1) (see Sect.8 )

that can improve the convergence of all used asymptotic procedures. The introducingof asymptotically small or large diffusivity (e.g. µ = O(1/ω) or µ = O(ω)) will generally

36 V. A. Vladimirov

lead to different asymptotic theories corresponding to three independent small parame-ters from the extended list (2.8).

12. In Sect.7.9 we have shown that the drift, caused by a relatively simple oscilla-tory velocity, produces chaotic dynamics of particles. This result leads to numerous newquestions (see the end of Sect.7.9 ) and deserves a serious elaboration.

13. In Sect.2.3 we have demonstrated that an infinite and continuous range of slow-time scales 1/ω∗ < T < ∞ can be used in an important case where slow-time t is absentin the expression for velocity: u = u(x, τ). We have formulated this result with the aimto clarify the existence of many scales (which often causes confusions) and to show thatan infinite number of similar (to the considered in Sects.3-7 ) solutions are available. Atthe same time, this result can lead to studying the motions with simultaneous presenceof several different scales T , and to developing multi-scale (triple-scale, etc.) theories.

14. It is worth completing the calculations of Riemann’s invariant and (8.64) and tocompare the related solutions for a with that of (4.25). The aim is to find the advantagesgiven by two complementary averaged solutions of the same problem.

15. The version of TTAM -theory for a vectorial passive admixture (with the av-eraged equations (8.69)) is linked with the problem of kinematic MHD-dynamo (seeMoffatt (1978)) and can bring new results. It is apparent, that for the majority of driftvelocities V (x, t) the stretching of material elements will produce linear growth |h| ∼ t.At the same time, there are known examples with an exponential stretching of mate-rial lines with time, say, in the flows near stagnation points or in chaotic flows. Theseexamples will provide the exponential growth of |h|.16. Our theory brings up a new possibility: one can find an oscillatory velocity u

which produces any required drift field (for example, a drift in the form of ABC-flow).The results of Sects.4-7 (including (7.7),(7.38)) indicate that the solution of this problemis not unique.

17. TTAM -method and results have potential applications in the studies of a broadvariety of flows. The discovery of drift motions with the velocities V 1 = O(1) andV 2 = O(1) can lead to new applications. The other advantages of our method, that can beexploited in applications, are: (i) the mathematical generality of the imposed oscillatoryvelocity fields (2.2),(6.1); (ii) the wide class of scalings (2.14); and (iii) the most straight-forward Eulerian average. In particular, our method can be useful for applications tobiologically motivated flows possessing complex geometry (e.g. Hydon & Pedley (1993),Magar & Pedley (2005), Leptos, Guasto, Gollub, Pesci, and Goldstein (2009)), for micro-hydrodynamics (e.g. Leal (1980), Kim & Karriba (1991), Hinch & Nitsche (1993), Moffatt (1996));for flow mixing (e.g. Ottino (1989), Carlsson, Sen, and Lofdahl (2005), Carlsson, Sen, and Lofdahl (2004),Leal (2007)), for the flow of blood and flows in lungs, see Pedley (1980), and for the pipeswith oscillating walls, see Carpenter & Pedley (2001). The spreading of ash from recentlyerupted Eyjaffjallajokull volcano has made the research on the general area of the trans-port of an admixture in a fluid (which can oscillate due to various reasons) even moreimportant.

This research was partially supported by EPSRC grants GR/S96616/01, GR/S96616/02,and EP/D035635/1. The author thanks the Department of Mathematics of the Univer-sity of York for the research-stimulating environment. The author is grateful to Profs.A.D.D.Craik, R.Grimshaw, K.I.Ilin, M.E.McIntyre, H.K.Moffatt, A.B.Morgulis, T.J.Pedley,and V.A.Zheligovsky for helpful discussions.

Admixture and Drift in Oscillating Fluid Flows 37

Appendix A. Calculations for α = 1/2 of Sect.4

The zero-order equation (4.5) is:

a0τ = 0 (A 1)

The substitution of a0 = a0(x, t) + a0(x, t, τ) into (A 1) gives a0τ = 0. Its T-integration(2.21) produces a unique (inside the T-class) solution a0 ≡ 0. At the same time (A 1)does not impose any restrictions on a0(x, t), which must be determined from the nextapproximations. Thus the results derivable from (A 1) are:

a0(x, t, τ) ≡ 0, ∀a0 = a0(x, t); a[0] ≡ a0 + a0 = a0(x, t) (A 2)

The substitution of the truncated solution a[0] into the governing equation (4.3) producesthe RHS-residual of order ε:

D2a[0] = Res[0] ≡ ε(u · ∇) a0 + ε2a0t = O(ε) (A 3)

The first-order equation (4.6) is

a1τ = −(u · ∇) a0 (A 4)

The use of a0 ≡ 0 (A 2) and a1τ ≡ 0 reduces (A 4) to the equation a1τ = −(u · ∇) a0. ItsT-integration (2.21) gives the unique solution for a1

a1 = −(ξ · ∇) a0 (A 5)

where ξ ≡ uτ (4.16). Hence, a1 and a[1] are

a1 = a1 − (ξ · ∇) a0, a[1] = a0 + εa1; ∀a0(x, t) and a1(x, t) (A 6)

The substitution of a[1] into (4.3) produces the RHS-residual of order ε2:

D2a[1] = Res[1] ≡ ε2

((u · ∇) a1 + ∂t a

[1])= O(ε2) (A 7)

Formulae (A 2) and (A5) give (4.11) and (4.12).The second-order equation ((4.7) for n = 2) is

a2τ = −(u · ∇) a1 − a0t (A 8)

The use of (A 2) and a2τ ≡ 0 transforms (A 8) to

a2τ = −(u · ∇) a1 − (u · ∇) a1 − a0t (A 9)

Its B-part is

a0t = −〈(u · ∇) a1〉 (A 10)

where we have used 〈a2τ 〉 = 0, 〈(u ·∇) a1〉 = 0, and 〈a0t〉 = a0t. The substitution of (A 5)into (A 10) produces the equation

a0t = 〈(u · ∇)(ξ · ∇)〉 a0 (A 11)

which represents a version of inspection equation (3.13) systematically derived. One mayexpect that the RHS of (A 11) contains both first and second spatial derivatives of a0,however all second derivatives vanish. In order to prove it we introduce a commutator(2.23)

K ≡ [ξ, u] = (u · ∇)ξ − (ξ · ∇)u, (A 12)

(u · ∇)(ξ · ∇)− (ξ · ∇)(u · ∇) = K · ∇. (A 13)

38 V. A. Vladimirov

The B-part of (A 13) is

〈(u · ∇)(ξ · ∇)〉 = 〈(ξ · ∇)(u · ∇)〉+K · ∇ (A 14)

At the same time the integration by parts (2.29) gives

〈(u · ∇)(ξ · ∇)〉 = −〈(ξ · ∇)(u · ∇)〉, u ≡ ξτ (A 15)

Combining (A 14) and (A15) we obtain

〈(u · ∇)(ξ · ∇)〉 = 1

2K · ∇ (A 16)

which reduces (A 11) to the advection equation (4.17) with

V 0 ≡ −〈(u · ∇) ξ〉 = −1

2〈[ξ, u]〉 = −1

2K (A 17)

which coincides with (4.20). The T-part of (A 9) appears after subtracting (A 10) from(A 9) and the use of notation (2.28):

a2τ = −(u · ∇) a1 − (u · ∇) a1. (A 18)

Its T-integration (2.21) with the use of (A 5) gives (4.13):

a2 = −(ξ · ∇) a1 + (u · ∇)(ξ · ∇)τa0, ∀a1 (A 19)

Hence, a2 and a[2] can be written as

a2 = a2 + a2, a[2] = a0 + ε(a1 − (ξ · ∇) a0

)+ ε2a2, ∀a1, a2 (A 20)

where a0 and a2 are given by (4.17), (A 19). The substitution of a[2] into (4.3) producesthe RHS-residual of order ε3

D2a[2] = Res[2] ≡ ε3 ((u · ∇) a2 + ∂t (a1 + εa2)) = O(ε3). (A 21)

The third-order equation ((4.7) for n = 3) is:

a3τ = −(u · ∇) a2 − a1t (A 22)

Its B-part is

a1t = −〈(u · ∇) a2〉. (A 23)

The substitution of (A 19) into (A 23), the use of u ≡ ξτ , and the integration by parts(2.29) yield

a1t = 〈(u · ∇)(ξ · ∇)〉a1 + 〈(ξ · ∇)(u · ∇)(ξ · ∇)〉a0 (A 24)

where 〈(u ·∇)(ξ ·∇)〉 has been already simplified in (A 16). The second term in the RHSof (A 24) formally contains the third, second, and first spatial derivatives of a0; howeverall the third and second derivatives vanish. To prove it, first, we use (2.30):

〈(ξ · ∇)(u · ∇)(ξ · ∇)〉 = −〈(u · ∇)(ξ · ∇)(ξ · ∇)〉 − 〈(ξ · ∇)(ξ · ∇)(u · ∇)〉 (A 25)

Then we use (A 12), (A 13) to transform the sequence of operators (ξ · ∇) and (u · ∇) ineach term in the RHS of (A 25) into their sequence in the LHS. The result is

〈(ξ · ∇)(u · ∇)(ξ · ∇)〉 = 1

3K′ · ∇, K ′ ≡ [K, ξ] (A 26)

Admixture and Drift in Oscillating Fluid Flows 39

As the result (A 24) takes form (4.18) with V 0 (A 17) and

V 1 ≡ −〈(ξ · ∇)(u · ∇)ξ)〉 = −1

3〈[[ξ, u], ξ]〉 = −1

3K′ (A 27)

which gives (4.20). The T-part of (A 22) after its T-integration gives (4.14)

a3 = −(ξ · ∇) a2 − (u · ∇) a2τ − aτ1t, aτ1 = −(ξτ · ∇) a0 (A 28)

where a2 is given by (A19).Hence, a3 and a[3] can be written as

a3 = a3 + a3, a[3] = a0 + ε(a1 − (ξ · ∇) a0) + ε2a2 + ε3a3, ∀a2, a3 (A 29)

where a0, a1, a2, and a3 are given by (4.17), (4.18), (A 19), and (A28). The substitutionof a[3] into (4.3) produces the RHS-residual of order ε4:

D2a[3] = Res[3] = O(ε4) (A 30)

Explicit formulae for residuals (similar to (A 3), (A 7), (A 21)) for (A 30), (4.29) and forall other considered cases can be calculated straightforwardly; however for brevity we donot present them.The fourth-order equation ((4.7) for n = 4) is:

a4τ = −(u · ∇) a3 − a2t (A 31)

Its B-part is

a2t = −〈(u · ∇) a3〉 (A 32)

The substitution of a3 (A 28) into (A 32), a2 (A 19) into a3 (A 28), the integration byparts (2.29), and the use of (4.16), (A 17), (A 27) yield

〈(u · ∇) a3〉 = (V 0 · ∇)a2 + (V 1 · ∇)a1 + 〈(ξ · ∇)(ξ · ∇)〉(V 0 · ∇)a0 − (A 33)

−〈(ξ · ∇)(ξt · ∇)〉a0 + Xa0, where X ≡ 〈(ξ · ∇)(u · ∇)(u · ∇)(ξ · ∇)τ 〉

The shorthand operator X (as well as the operators Y, A, B, C, and F below) actson a0. The RHS of (A 33) formally contains the fourth, third, second, and first spatialderivatives of a0; however all the fourth and third derivatives vanish. In order to prove itwe first rewrite X as

X = 〈(ξ · ∇)(u · ∇)Yτ 〉 where Y ≡ (u · ∇)(ξ · ∇) (A 34)

The use of (2.30) and (A 12), (A 13) transforms (A 34) to

2X = −A+B+1

2〈(ξ · ∇)(ξ · ∇)〉(K · ∇) (A 35)

A ≡ 〈(ξ · ∇)(ξ · ∇)(u · ∇)(ξ · ∇)〉, B ≡ 〈(Kτ· ∇)(u · ∇)(ξ · ∇)〉

Let us now simplify A and B. For B we use (2.30)

B ≡ 〈(Kτ· ∇)(u · ∇)(ξ · ∇)〉 = −〈(K · ∇)(ξ · ∇)(ξ · ∇)〉 − 〈(K

τ· ∇)(ξ · ∇)(u · ∇)〉

To change (ξ · ∇)(u · ∇) into (u · ∇)(ξ · ∇) in the last term we use (A 12), (A 13) thatyields:

B = −1

2〈(K · ∇)(ξ · ∇)(ξ · ∇)〉 − 1

4κ · ∇, κ ≡ [K

τ, K] (A 36)

40 V. A. Vladimirov

The operator A is simplified by the version of (2.30) with four multipliers

A ≡ 〈(ξ · ∇)(ξ · ∇)(u · ∇)(ξ · ∇)〉 = −〈(u · ∇)(ξ · ∇)(ξ · ∇)(ξ · ∇)〉 − (A 37)

−〈(ξ · ∇)(u · ∇)(ξ · ∇)(ξ · ∇)〉 − 〈(ξ · ∇)(ξ · ∇)(ξ · ∇)(u · ∇)〉The multiple use of commutator (A 12), (A 13) allows us to transform the sequence of

operators (ξ · ∇) and (u · ∇) in each term in the RHS of (A 37) to the sequence in itsLHS. The result is

A = −1

2〈(K · ∇)(ξ · ∇)(ξ · ∇)〉+ 1

4K ′′ · ∇, K ′′ ≡ [K′, ξ] (A 38)

Now, (A 35), (A 36), and (A38) yield

X =1

4(K · ∇)〈(ξ · ∇)(ξ · ∇)〉+ 1

4〈(ξ · ∇)(ξ · ∇)〉(K · ∇)− 1

8(κ+K ′′) · ∇

The substitution of this expression into (A 33), (A 32) gives

a2t + (V 0 · ∇)a2 + (V 1 · ∇)a1 −1

8(κ+K ′′) · ∇a0 +

1

4Ca0 − Fa0 = 0, (A 39)

C ≡ (K · ∇)〈(ξ · ∇)(ξ · ∇)〉 − 〈(ξ · ∇)(ξ · ∇)〉(K · ∇), F ≡ 〈(ξ · ∇)(ξt · ∇)〉Additional transformations of the last two operators in (A 39) yield

1

4C− F =

1

2〈[ξ, ξt]〉 · ∇ − 1

2〈(u′ · ∇)ξ + (ξ · ∇)u′〉 · ∇ − 1

2〈u′

iξk + u′

k ξi〉∂2

∂xi∂xk=

=1

2〈[ξ, ξt]〉 · ∇ − ∂

∂xk

(χik

∂

∂xi

)+

1

2〈ξdiv u′ + u

′div ξ〉 (A 40)

u′ ≡ ξt − [V 0, ξ], χik ≡ 1

2〈u′

iξk + u′

k ξi〉 (A 41)

The substitution of (A 40) into (A 39) leads to the equation for a2 (4.19) where theformula (4.23) for χik is obtained from (A41) by the use of definition u

′.The T-part of (A 31) after its T-integration gives (4.15)

a4 = −(ξ · ∇) a3 − (u · ∇) a3τ − aτ2t (A 42)

where a3 is given by (A28).

Appendix B. Calculations for Super-critical Families

B.1. Calculations for α = 1/3 of Sect.5

The equations of the zeroth and first approximations (5.5),(5.6) are the same as (4.5),(4.6)hence they have the same solutions

a0 = a0(x, t), a1 = a1 − (ξ · ∇)a0; ∀ a0, a1 ∈ B ∩O(1) (B 1)

An essential difference from the ω1/2-procedure emerges for the second approximation(5.7) where instead of an advection equation for a0 (4.17) we obtain a compatibilitycondition

(V 0 · ∇) a0 = 0 (B 2)

which represents an identity due to (5.1). The T-part of (5.7) produces the same a2 asin (A 19). Hence

a2 = a2 + a2, a2 = −(ξ · ∇) a1 + (u · ∇)(ξ · ∇)τa0, ∀a2 ∈ B ∩O(1) (B 3)

Admixture and Drift in Oscillating Fluid Flows 41

The equation for the third approximation (n = 3 in (5.8)) jointly with (5.1) produces theB-part of the equation

a0t = 〈(ξ · ∇)(u · ∇)(ξ · ∇)〉a0 (B 4)

with the same triple-correlation as in (A 26), (A 27); hence(

∂

∂t+ V 1 · ∇

)a0 = 0, where V 1 = −1

3〈[[ξ, u], ξ]〉 (B 5)

The T-part of (5.8) after its T-integration gives a simpler expression

a3 = −(ξ · ∇) a2 − (u · ∇) a2τ (B 6)

with a2 (B 3). Hence, a3 and a[3] can be written as

a3 = a3 + a3, a[3] = a0 + ε(a1 − (ξ · ∇) a0) + ε2a2 + ε3a3 (B 7)

where a0, a2, and a3 are given by (B 5), (B 3), and (B 6), while a1, a2 and a3 are to befound from further approximations. Hence the ω1/3-procedure shows that in the presenceof degeneration (5.1) a drift velocity remainsO(1) and is given by the expression V 1 (B 5).The equation for the fourth approximation ((5.8) with n = 4) is:

a4τ = −(u · ∇) a3 − a1t (B 8)

After the use of the same steps as in the ω1/2-procedure we have B-part

a1t + (V 1 · ∇)a1 + (V 2 · ∇)a0 = 0, (B 9)

with the same notations as in (B 5), (5.17), (5.18). The T-part of (B 8) after its T-integration gives

a4 = −(ξ · ∇) a3 − (u · ∇) a3τ − aτ1t (B 10)

with a3 (B 6).

B.2. Calculations for α = 1/4 of Sect.5

The equation of the zeroth, first, and second approximations (5.24),(5.25), and (5.26) arethe same as (5.5),(5.6), and (5.7) so they have the same solutions (B 1)-(B 3)

a0 ≡ a0 + a0 = a0(x, t), a1 = a1 − (ξ · ∇) a0, a2 = a2 + a2 (B 11)

An essential difference from the ω1/3-procedure emerges for the third approximation(5.26) where instead of an advection equation for a0 (B 5) we obtain a compatibilitycondition

(V 1 · ∇) a0 = 0 (B 12)

which represents an identity due to (5.20). The T-part of (5.27) produces the same a3 asin (B 6). The equation for the fourth approximation ((5.28) with n = 4) is:

a4τ = −(u · ∇) a3 − a0t (B 13)

After performing the same steps as in the ω1/3-procedure we obtain B-part

a0t + (V 2 · ∇)a0 = 0, (B 14)

with the same notations as in (5.17), (5.18). The T-part of (B 13) after its T-integrationgives

a4 = −(ξ · ∇) a3 − (u · ∇) a3τ (B 15)

42 V. A. Vladimirov

where a3 is given in (B 6).

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